A Collection of Matrices
for Testing Computational Algorithms

A Collection of Matrices
for Testing Computational Algorithms

To Alston S. Householder

A Collection of Matrices
for Testing Computational Algorithms
ROBERT T. GREGORY, Ph.D.
Professor of Mathematics and of Computer Sciences
Senior Research Mathematician, Computation Center
The University of Texas at Austin
DAVID L. KARNEY, M.A.
Department of Computer Sciences
The University of Texas at Austin
<£>
ROBERT E. KRIEGER PUBLISHING COMPANY
HUNTINGTON, NEW YORK „
1978

Original Edition 1969
Reprint 1978 with corrections
Printed and Published by
ROBERT E. KRIEGER PUBLISHING CO., INC.
645 NEW YORK AVENUE
HUNTINGTON, NEW YORK 11743
Copyright © 1969 by
JOHN WILEY & SONS, INC.
Reprinted by Arrangement
All rights reserved. No reproduction in any form of this
book, in whole or in part (except for brief quotation in
critical articles or reviews), may be made without written
authorization from the publisher.
Printed in the United States of America
Library of Congress Cataloging in Publication Data
Gregory, Robert Todd, 1920-
A collection of matrices for testing computational algorithms.
Reprint, with corrections, of the edition published by Wiley-
Interscience, New York.
Bibliography: p.
Includes index.
1. Matrices. 2. Algorithms. 3. Numerical analysis-
Data processing. I. Karney, David L., joint author.
II. Title.
[QA188.G72 1978] 512.9'43 77-19262
ISBN 0-88275-649-4

PREFACE
This monograph is intended primarily as a reference book for numerical
analysts and others who are interested in computational methods for solving
problems in matrix algebra. It is well known that a good mathematical algorithm
may or may not be a good computational algorithm. Consequently, what is needed
is a collection of numerical examples with which to test each algorithm as soon
as it is proposed. It is our hope that the matrices we have collected will help
fulfill this need.
The test matrices in this collection were obtained for the most part
by searching the current literature. However, four individuals who had begun
collections of their own contributed greatly to this effort by providing a large
number of test matrices at one time.
First, Joseph Elliott1s Master!s thesis [18] provided a large collection
of tridiagonal matrices. Second, Mrs. Susan Voigt, of the Naval Ship Research
and Development Center, contributed a varied collection of matrices. Third,
Professor Robert E. Greenwood, of The University of Texas at Austin, provided
a valuable list of references along with his collection of matrices and determinants.
Finally, just as this work was nearing completion, the collection of Dr. Joan
Westlake [60] was discovered. Her collection of 41 test matrices contained seven
which we had overlooked; therefore, they were added.
Matrix 6.11 is a non-Hermitian matrix of order 20. It is a specific
example of a class of matrices known as Dolph-Lewis matrices [14] which arose
around 1957 in an investigation of perturbations of plane Poiseuille flow.
Accurate eigenvalues, along with left and right eigenvectors and condition numbers,
were provided by Dr. J. H. Wilkinson of the National Physical Laboratory.

Matrix 3.8 is the finite segment (of order n) of the (infinite) Hilbert
Matrix. Matrix 3.26 is a generalization. The exact inverses of the finite
Hilbert segments exhibited were provided by Dr. Max Engeli of FIDES Treuhand-
Vereinigung, Zurich. Dr. Engeli-s program for computing these inverses was
written in SYMBAL, a language of his own creation.
The first author is grateful to Dr. Engeli and to Dr. Erwin Nievergelt
for making the facilities of FIDES, including the CDC 6500 computer, available
to him during his stay in Zurich.
Partial support for this work was provided by the National Science
Foundation under Grant GP 8442 and by the Army Research Office (Durham) under
Grant DA-ARO(D)-31-124-G721, at The University of Texas at Austin. This support
is gratefully acknowledged. We are also grateful to Professor George E. Forsythe,
who read the manuscript and offered many helpful suggestions.
The book is dedicated to Dr. Alston S. Householder, who has inspired
numerical analysts for the past two decades.
We are indebted to Mrs. Dorothy Baker for preparing the manuscript.
Her superb job of typing this difficult material enabled the publishers to use
photographic reproduction. This saved the authors an enormous amount of
additional proofreading and avoided the introduction of countless additional errors.
ROBERT T. GREGORY
DAVID L. KARNEY
AUSTIN, TEXAS
April 1969

TABLE OF CONTENTS
CHAPTER PAGE
I. INTRODUCTION 1
II. CONSTRUCTION OF TEST MATRICES 5
III. TEST MATRICES: INVERSES, SYSTEMS OF LINEAR EQUATIONS,
AND DETERMINANTS 29
IV. TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF REAL SYMMETRIC MATRICES 55
V. TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF REAL NONSYMMETRIC MATRICES 81
VI. TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF COMPLEX MATRICES 114
VII. TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF TRIDIAGONAL MATRICES 134
REFERENCES 143
SYMBOL TABLE 148
INDEX 151

CHAPTER I
INTRODUCTION
In order to test the accuracy of computer programs for solving
numerical problems, one needs numerical examples with known solutions. The
aim of this monograph is to provide the reader with suitable examples for
testing algorithms for finding the inverses, eigenvalues, and eigenvectors
of matrices. A collection of methods for constructing test matrices and a
large collection of numerical examples have been included. We have
endeavored to allow the reader much freedom in his choice of a test matrix.
Chapter II of this monograph describes methods for generating;
matrices with known inverses and eigensystems whereas Chapter III contains
test matrices with known inverses and solutions of systems of linear
equations.
In the later chapters test matrices with known eigenvalues and
eigenvectors are given. We have included, when possible, both right and left
eigenvectors. The reader is reminded that if A is an Hermitian matrix,
the left eigenvectors of A are the conjugate transpose of the right
eigenvectors. For some of the examples, the tridiagonal forms are given
which arise in the use of certain well-known algorithms for computing
eigenvalues. The methods of Givens and Householder, for example, transform real
symmetric matrices into the tridiagonal form

2 Matrices for Testing Computational Algorithms
ai *2
P2 a2 p3
P i a i P
^n-1 n-1 ^n
p a
n n
The method of Lanczos transforms nonsymmetric matrices into the tridiagonal
form
ai P2
1 a2 P3
n-1 Kn
a
n
The examples exhibited in this monograph include both well-conditioned
and ill-conditioned matrices. For each example we have computed several
condition numbers, and for the ill-conditioned matrices the condition numbers
are included.
Let A = [a..] be an n x n nonsingular matrix with eigenvalues
X , X , ..., X . For the problem of matrix inversion, at least three
condition numbers are used. Von Neumann and Goldstine [59] suggest the condition
number
P(A)
max X.I
1 i'
l
min I X. I

Introduction 3
Turing [57] proposes the two condition numbers
M(A) = n max |a. . | max |a. . |
i,j 1J i,j 1J
and
N(A) -jj llAl^-llA-1!^ ,
where
IIaIL -
n n
i=l j=l ^
and where
A"1 = [a..].
It can be shown that P(A) and N(A) do not differ very much from M(A) .
In particular, we have [60, p. 90] , [53]
— M(A) ^ N(A) ^ M(A)
n
and
P(A) ^ nM(A).
If A is symmetric, we also have
- M(A) ^ P(A).
If the matrix elements are chosen at random from a normal population, then
an N-condition number of order vn and an M-condition number of order yn log n
can be expected.
Actually, M(A) and N(A) are not used as much as the more general
condition number
k(a) = HaIHIa"1!!,

4 Matrices for Testing Computational Algorithms
for various norms, not necessarily the same.
Now let x^ and y^ ' be, respectively, right and left eigenvectors
of A corresponding to the eigenvalue X., and suppose x^ ' and y^ ' are
normalized so that
n ,.v rt n
SlxW-ZlyWl2-!.
j=l j j=l j
The condition of A with respect to the eigenvalue problem can be measured by
the n condition numbers of A [62, pp. 88-89], |s.| , where
i
s^y^x^, i= 1,2 n.
Here, |s.| is the condition number for A.. Thus, some eigenvalues may be more
ill-conditioned than others. Observe that if A is Hermitian, we have, for
all i,
s. = 1.
i

CHAPTER II
CONSTRUCTION OF TEST MATRICES
1. In this chapter we present a variety of methods by which the
reader can construct matrices with known inverses, eigenvalues, and
eigenvectors. We begin with the following well-known results which can be found
in elementary texts on matrix algebra such as Hohn [28].
T
Theorem 1. The eigenvalues of A and A are the same.
— H
Theorem 2. The eigenvalues of A and A are the conjugates of the
eigenvalues of A.
Theorem 3. The eigenvalues of A are the reciprocals of the eigenvalues
of A.
Theorem 4. If X , \ > ..., X are the eigenvalues of an n x n matrix A
and if P(a) is a polynomial, then the eigenvalues of P(A) are P(A-), P(A ), ...,
P(A ). Further, if x is an eigenvector of A corresponding to the eigenvalue A,
then x is an eigenvector of P(A) corresponding to P(A).
Theorem 5. The matrix
A =
al
1
0
0
-a2 .
0
1
0
-a .
n-1
0
0
1
-a
n
0
0
0
5

6 Matrices for Testing Computational Algorithms
has the characteristic equation
I A-All = An + a, A11"1 + ... + a = 0.
1 ' 1 n
Theorem 6. If B is a non-singular matrix, then the eigenvalues of
BAB are the same as those of A. If x and y are,respectively, right and left
eigenvectors of A corresponding to the eigenvalue A, then Bx and yB are
respectively right and left eigenvectors of BAB corresponding to the
eigenvalue A. If A is also non-singular, then (BAB ) = BA B .
2. One of the simplest methods of constructing test matrices is by
forming composite matrices (some authors use compound matrices). In this
regard we have the following.
Theorem 7.[2]. The eigenvalues of a block-diagonal matrix,
diag [A-, A , ..., A- ] , are the eigenvalues of A-, A , ..., A,.
Theorem 8 [28, pp. 81-82]. Suppose B is composed of submatrices of
indicated orders,
B =
All
(n x n)
A21
(m x n)
■"i"
A12
(n x m)
A
A22
(m x m)
-1
and §upps§§ A** and P = A** - -21^A11A12^ are non"sin8ular- Then B is non"
singular, and if we partition B into submatrices

Construction of Test Matrices 7
-i
Bll
(n x n)
B21
(m x n)
i Bi2
J (n x m)
-I.. _-----_
j B22
[ (m x m)
we have
11
12
>21
^l^iV"1^
-<AUA12>P"1
-p"1<A2iAn)
B22 = P
Theorem 9 [36, p. 12]. If A and B are real n x n matrices and
S =
A B
B A
then the eigenvalues of S are the eigenvalues of A + B together with the
eigenvalues of A - B.
Another class of composite matrices suitable for test purposes can
be obtained by the use of Kronecker products. Most of the following material
comes from Bellman [2, Chapter 12] and Marcus [36]; the reader is referred
to Friedman [25] for additional information.
Definition, [2] . Let A = [a. .] be an m x m matrix and B an n x n
matrix. The mn x mn matrix defined by

8 Matrices for Testing Computational Algorithms
allB a12B
a21B a22B
a .B a B
ml vol
a. B
lm
a2mB
a B
mm
is called the Kronecker product of A and B and is denoted by A (§) B,
For matrices of this form we have the following very important results,
Theorem 10 [2]. If A is an m x m matrix with eigenvalues A., i =
1, 2, ..., m, and B is an n x n matrix with eigenvalues jj.., j = 1, 2, . .., n,
then the eigenvalues of A <§ B are A.jj.., i = 1, 2, . .., m and j = 1, 2, ..., n.
The eigenvectors are mn x 1 column-vectors of the form
Jij
x<i)y(j)
x(i)y(j)
m J
where yr J is an eigenvector of B corresponding to the eigenvalue p.. and x£ ,
J k
k = 1, 2, .. . , m, denote the components of the eigenvector x of A
corresponding to A. .
Theorem 11 [36, p. 5]. If A and B are non-singular, then A ® B is
non-singular and (A ® B) = A <g) B
We can, of course, consider Kronecker powers of a particular matrix,
i.e.
A(2) = A® A
,<k+1> = A ® A(k> .

,(k)
Construction of Test Matrices 9
The eigenvalues of Avw are all possible products consisting of k factors,
each of which is an eigenvalue of A. We can also define matrices having
eigenvalues of this form which are of much smaller dimension than the general Kron-
ecker product. For example [2], suppose
A =
all a12
a21 a22
and suppose A has eigenvalues A-, A-. Starting with the equations
A1X1 " allXl + a12X2
A1X2 " a21Xl + a22X2 *
we form the products, for a fixed integer k,
k-i i
^allXl + al2X2^ ^a21Xl + a22X2^ ' i = 0, 1, ..., k.
Then, if we let A.- v = [b .], i,j = 0, 1, ..., k, denote the k+1 x k+1 matrix
k-i i
such that b. . is the coefficient of the x- Jx;J term in the product
ij 12^
k-i i k-i i
^allXl + a12X2^ ^a21Xl + a22X2^ ' the eiSenvalues of A/k\ are \ \>
1 ~ U, J., ■•■, Jv ■
For example, if k = 2, we have the products
, , \2 _ 2 2 , . j.22
CallXl + a12X2} " allXl + 2allal2XlX2 + a12X2
(allXl + a12X2)(a21Xl + a22X2> = aUa21Xl + (alla22 + a12a21)xlV a12&22X2
i x ^ - 2 2 4. o j.22
U21X1 a22X2; " a21Xl Za21a22XlX2 a22X2 '
Thus the matrix

10 Matrices for Testing Computational Algorithms
\2)
r 2
alla21
2
a21
2alla12
(aUa22+ a12a21)
2a2la22
2 1
a12
S12a22
2
a22
2 2
has eigenvalues A-, A-A , A~ .
Next let us suppose that A is an m x m matrix with eigenvalues A.,
i = 1, 2, . .., m, and B is an n x n matrix with eigenvalues |i., j = 1, 2, . .., n.
It can be shown that the mn eigenvalues of (I ® B) + ( A ® I ) are A.+ jj..,
for all i and j. For example, let A be the m x m matrix
A =
a b
b a b
b a b
b a b
b a
and let B be the n x n matrix
B =
d c
c d c
c d c
c d c
c d
k/r
where the eigenvalues of A are a + 2b cos —- , k * 1, 2, - - -, m, and those
k/r
of B are d + 2c cos —- , k ~ 1, 2, . . ., n. Then the eigenvalues of

Construction of Test Matrices 11
(Im ® B) + ( A ® In) =
(al + B)
n
bl
n
bl
n
(al + B)
n
bl
n
bl
n
(al + B) bl
v n n
bl (al + B)
n n
are given by
A. . = a + d + 2b cos ~7 + 2c cos -^7 ,
lj m+1 n+1 '
i - 1, 2, ■ ■■, ni; j = 1, 2, ---, n.
3. Determinants can also be used to define another type of matrix
"power.11 For simplicity we shall consider a 3 x 3 matrix A and a set of 2 x 2
determinants formed from the eigenvectors of A. The procedure can be
generalized to treat m x m determinants associated with the eigenvectors of n x n
matrices [2].
Let
A =
al a2
C2 C3
and let the eigenvalues of A be A., A~, A„ with associated eigenvectors x
(2) (3)
x , x . Now define, for arbitrary n-dimensional vectors r and s,
(1)
8ij(r's) =det
ri si
rj SJ
9 ^> J ~" *•» ^ y ■
n.

12 Matrices for Testing Computational Algorithms
Then, for our example, it can be shown that the matrix
G =
g12(a'b> g23(a.t>) g31(a,b)
g12(a,c) g23*a,C* §31(a»c)
gio(b,c) g„(b,c) g^(b,c)
'12
'23
31'
has eigenvalues A.A2, A.A , A„A_. Also, corresponding to the eigenvalue A.A.,
i ^ j, there is an eigenvector
y =
where
7l = g12(x(i>,x<J>), y2 = g23(x<i>,x^>), and y3 = g31(x(1> ,x^).
4. Brenner [7] has described another set of composite matrices which
can be used to test inversion and eigenvalue routines. Let f denote the
n x 1 column-vector whose components are all lfs. For arbitrary integers n
T
and k, let J , = f f. , i.e., J , is the n x k matrix whose elements are all
' nk n k* ' nk
lfs. The matrix J has the following properties; f is an eigenvector of
J corresponding to the eigenvalue A = n; every vector orthogonal to f is
an eigenvector of J corresponding to the eigenvalue A = 0. The eigenvalue
A = 0 has multiplicity n - 1, and its associated invariant space is spanned
by the vectors g. = f - n e , i = 1, 2, ..., n-1, where e is the n x 1 column
vector which has components 6.., j ~ 1, 2, ..., n. This leads us to the
following result.

Theorem 13. The matrix
Construction of Test Matrices 13
A =
(a.I +b..J )
1 nl U Vl
b21JVl
b„J
12 V2
(a,I + b„J )
2 n2 22 nun
b-iJ
tl n^
bt2Jntn2
bn-J
It Vt
b9l-J
2t n2nt
(a I + b J )
t nt tt ntnt
is similar to the block-diagonal matrix
diag[Ar A2, ..., Afc, A +1]
where, for i = 1, 2. .... t, A. = a.I -, and A^,- is the t x t matrix defined
9 l i n -1 t+1
by
t+1
(a1+ bnn1)
b21nl
b12n2
(a2"*22n2) '••
bltnt
b2tnt
btlnl bt2n2
(afc+ bttnt)
For r = 1, 2, ..., t, the vectors
vir
f - n e
n r n
r r
i - 1, 2,
.., n - lf
' r

14 Matrices for Testing Computational Algorithms
are the eigenvectors of A corresponding to the eigenvalue A = a , which has
multiplicity n - 1.
The determinants and inverses can also be obtained for matrices of
this form. We illustrate with the example.
B =
(al + bJ )
n nn'
cJ
dJ
mn
nm
(hi + w >
m mm
The matrix B is similar to the matrix diag[A ,A ,A ] where
A. = al
n-1
Art = hi
A. =
tn-1
(a+bn) cm
dn (h+km)
From this it follows that the eigenvalues of B are a, h, and the eigenvalues
of A«. Also, the determinant of B is seen to be a h [(a+bn)(h+km)-cdnm],
Writing A in the form
(a"V bfn)
d'n (h + k'm)
where
i - h+km-a A
b1 =
1 - - —
An
c =
d1 = - — kf =
a A ' K
i - a+bn-h A
Am
A = (a+bn)(h+km) - cdnm,
produces the inverse

Construction of Test Matrices 15
«-l
B =
(a"
'lI + b'J )
n nn
d'J
mn
C*J
nm
(1x1 + k'J )
m mm
5. The next method we shall discuss is due to Newberry [41].
Consider a matrix of the form
Q =
S R
C D
where S is a scalar, R is a row-vector [r_, r_, ..., r ], C is a column-
vector [c„, c_, ..., c ]T} and D is a diagonal matrix with diagonal elements
d„, d_, .... d . The inverse can be written in the form
2 3 ' n
.-1
S'
C
R'
M'
where each submatrix of Q has the same form as the corresponding submatrix
of Q except that M' is, in general, not diagonal. It can be shown that
• - (s - j2 Vi4l
and, for i,j = 2, 3, ..., n,
ci = -s,ci/di
rj = "S'^A^
V <V Vj>/di
Let A be an eigenvalue of Q, and let

16 Matrices for Testing Computational Algorithms
1
X,,
X =
X
n
be an associated eigenvector. Then the equation Qx
ing set of n equations:
Ax leads to the follow-
n
S + £ r x. » A
i«2
Ci + diXi = ^Xi* i s 2, 3, ..,, n.
Eliminating the x. yields
(i)
n
S + E r c /(A-d ) - A * 0.
i«2 x x x
If we write
and
n
P(A) = TT (A-d.)
i=2 x
Pi(A) = P(A)/(A-di), i = 2, 3, ..., n,
then (1) can be written
(A-S)P(A) - Z r.c P (A) = 0.
i=2 x x x
This is the characteristic equation of Q, and the following statements can be
made concerning the eigenvalues:
(a) If all re. > 0 and all d. are distinct, then all the eigenvalues
are real and are separated by the d,.

Construction of Test Matrices 17
(b) If all d. are equal to d, then d is an eigenvalue of multiplicity
n - 2. The remaining two eigenvalues are the zeros of
(2)
A - (S-M)A + Sd - Z re
i=2 1 l
and are real if, and only if,
n
(S-d) + 4 Z re > 0.
i=2 X X
(c) If all d. are equal to d, the eigenvectors associated with the
multiple eigenvalue d have zero as their first component and are orthogonal
to the vector [0, r , r_, ..., r ]. If A is a zero of (2), the eigenvector
corresponding to A is
A - d
P
6. Cline [13] also describes a general class of matrices with complex
elements for which the inverse, eigenvalues, and eigenvectors are known. Let
k be any real number such that k ^ -1, Let I be the identity matrix of order n,
and let B be any matrix with complex elements having n columns. Suppose
further that B has orthonormal rows. Then it follows that
(I+kB1^)"1 = I - -££ ih.

18 Matrices for Testing Computational Algorithms
Since
(I+kBHB)BH = (l+k)BH
and
(I+kBHB)(I-BHB) = I - BHB,
IT
it follows that the columns of B provide an orthonormal set of eigenvectors
it
of (I+kB B) corresponding to the eigenvalue A = 1 + k and that the columns
II
of (I-B B) contain a linearly independent set of eigenvectors corresponding
H H
to the eigenvalue A = 1. Now the rank of (I+kB B) is equal to the rank of B
H H
plus the rank of (I-B B). Thus B and any maximal linearly independent set
ji
of columns of (I-B B) form a complete set of eigenvectors. It should also be
II
pointed out that (I+kB B) is Hermitian.
By taking B as the 1 x n matrix [n , n , ..., n ] and k = n/(d-l)
where d ^ 1 and d ^ -(n-1), we can obtain the test matrix of Pei [46]:
i. ct ]
where
fcu= <
d, if i = j
1, if i * j.
We can write
and
T = (d-l)I + nBHB
= (d-l)(I+kBHB)
T"1 - -i- (T- JS_ RH1V>
T " d-1 (I k+1 B B)
Ti«-l±i**»-
Thus, if T* = [s. J, we have

Construction of Test Matrices 19
f
sij
d+n-2
= <
^
d(d+n-2)-(n-1)
-I
d(d+n-2)-(n-l)
, if i = j
, if i * J.
Furthermore, the eigenvalues of T are
A = (d-l)(l+k)
= n + d - 1
of multiplicity one and A = d - 1 of multiplicity n-1. Also,
l_
x/3
[l
1
1
1
* *
[1_
1
» v/T
1
-1
0
0
•
•
_0_
1
' v/S
ll
1
-2
0
• 1
• 1
OJ
Wn^ry
i
(n-1)
form an orthonormal set of eigenvectors of T, where the first corresponds to
the eigenvalue A = n + d - 1.
7. Ortega [44] describes a valuable method using similarity trans-
formations to generate test matrices. Let C - I + uv where u and v are
n x 1 column-vectors. Then
(f1 - I - (l-W^rW1.
It can be shown that any vector orthogonal to v is an eigenvector of C
corresponding to the eigenvalue A = 1 and that u. is an eigenvector of C cor-
ij
responding to the eigenvalue A = 1 + v u. Since the eigenvalue A =* 1 has
multiplicity n-1, the matrices C have limited use in testing eigenvalue
routines. They are quite useful, however, in testing inversion procedures.

20 Matrices for Testing Computational Algorithms
H -1
Now let a = (1+v u) , and let R be any n x n matrix. Then the
similarity transformation A = CRC becomes
A = (I-hiv^RCl+uvV1
= R + uvR - aRuv - a(v Ku)uv .
The inverse is given by
A"1 = CR'V1
-1 ^ IL-l -1 H , H-l . H
= R +uvR -aRuv- a(v K u)uv .
-1
Proper selection of u, v, and R will insure that A and A can be generated
exactly in the computer. To illustrate the possibilities, we present the
following examples. For simplicity we consider only real u, v, and R unless
stated otherwise.
Real symmetric matrices can be generated by letting
n
u = -2v, Z v. = 1
i=l X
R = D = diagtd^ d^ ..., d^.
T
Then (I-2w ) is orthogonal, and
A = (I-2wT)D(I-2wT)
= D - 2wTD - 2DwT + 4(vTDv)vvT.
The matrix A is symmetric, has eigenvalues d-, d0, ..., d , and eigenvectors
JL Z n
T
which are the columns of (I-2w ). In particular, if A = [a. .] and
v = [n % n % ..., n *],
then

Construction of Test Matrices 21
atJ = n-^nd^j- 2^- 2dj+ 2r)
where
1 n
r = 2n E d. .
k=l k
If we let R = diag[R-, R~, ..., R ] be a block-diagonal matrix such
that R. is a complex Hermitian matrix, then
A = (I-2wT)R(I-2wT)
will also be Hermitian.
To generate nonsymmetric real matrices, we have a much wider choice
T
for u and v, although the restriction u v = 0 affords some simplification.
T T
For example, if n is even, say n = 2k, u = c[l,l,...,1], v - [l,l,...,l,-l,-l,
T
..., -1] with k components of 1 and k components of -1, and a = v Du, then
T T T
A = D + uv D - Duv - auv .
It can be shown that, if A = [a..],
di5ij " C(di" V a)' i ^ j ^ k
aij =
V
di8ij + c(di" dj+ a)' k+1 ^ J ^ n-
The matrix A has real eigenvalues d-, d , ..., d , right eigenvectors which
T
are the columns of (I+uv ), and left eigenvectors which are the rows of
T
(I-uv ). It is easy to generate A exactly since only additions of the d
and multiplications by c are involved; if c - 1, only additions are required.
The parameter c provides some control over the condition of the problem since
the n condition numbers of A are

22 Matrices for Testing Computational Algorithms
{[(l+c)2 + (n-l)c2][(l-c)2 + (n-l)c2]}"% .
T
Another choice for u and v is u = [l, 2, ..., k, 1, 2, ..., k] and
T T
v = [l, 2, ..., k, -1, -2, ..., -k]. The relation u v = 0 is maintained,
and the n condition numbers are
sm = sk4m = {[l+2m2(3+l)][l+2m2(p-l)]f\
k 2
where m = 1, 2, ..., k, and p = E i .
i=l
Now if we let R = diag[R.. , R„ R ] be a block-diagonal matrix,
we can obtain a real matrix A which has complex eigenvalues. For example,
the R may be 2 x 2 real matrices which have known complex eigenvalues.
8. Next we consider a family of matrices, called circulants, which
are of the form
C =
C0 Cl C2
Vi co ci
Cn-2 Cn-1 C0
Cl C2 C3
"n-1
"n-2
"n-3
Let r, = exp , k = 1, 2, ..., n, be the solutions of the equation r = 1,
k. n
Then it can be shown [2, pp. 234-2351 that, for k = 1, 2, ..., n,
*k = C0 + Clrk + C2rk +
n-1
• + c -r-
n-1 k
is an eigenvalue of C with associated right eigenvector

Construction of Test Matrices 23
,<k>-
n-l
and left eigenvector
,(k) _ fr11"1 r11"2 r ll
f ~~ \a ' k * *# * * k*
Observe that since the r, are all distinct, C has n distinct right eigenvectors
and n distinct left eigenvectors. Note also that if we choose c. = c ., the
° i n-i'
matrix C is symmetric. Circulants can be generalized [2, p. 235] by using
equations of the form
n , n-1 , , n-2 , , ,
r = b-r + b0r + . .. + b .
12 n
9. Brenner [6] has defined a set of matrices related to the Mahler
matrices [35] for which the determinant, eigenvalues, eigenvectors, and
elementary divisors are known. Let k and n be positive integers such that
(k,n) =1, k > 1, n > 1, and let m be a positive integer with (m,n) = 1,
2?ri
m = 1 (mod k). Set s = exp -^r— , and let Q be the n x n matrix defined by
0 10
0 0 1
0 0 0
s 0 0
0 0
0 0
0 1
0 0

24 Matrices for Testing Computational Algorithms
Define v to be the 1 x n row-vector
* Li, 1, • • •, ij •
Finally let A(m) denote the n x n matrix defined by
A(m) -
v(I-Q )
v(Q -Q )
v(Q(n-l)m_Qnm)
where the r-th row of A(m) is v(Q - Q ), r = 1, 2, ..., n. It should
be pointed out that if k = 2, the elements of A(m) are 1, 0, -1.
It can be shown that there exists a non-singular matrix M such that
M A(m)M is a block-diagonal matrix. To obtain M and the block-diagonal form,
we need the following definitions. Let
CD(t) = exp[27ri(kt-k+l)/kn].
Define the function gt mod n of t mod n by
gt = mt + (k-l)[(m-l)/k] (mod n).
It follows that, if r is a positive integer,
grt = mrt + (k-l)[(mr-l)/k] (mod n),
Next let w(t,A(m)) be the function defined by
m.
w(t,A(„)> - i^fi

Construction of Test Matrices 25
Now define an equivalence relation f,~fl on the residue classes mod n by t- ~ t~
if, and only if, t- a g t for some positive integer r. Let t-, t , ..., t
be representative members of the equivalence classes of the relation l,~," and
let n., j =» 1, 2, ..., p, be the number of elements in the j-th equivalence
'j ~ j'
Let x(t) be the n x 1 column-vector given by
x(t) =
a>(t)
o>2(t)
(U (t)
, t* jl , Cm j m • ■, n#
Define the matrix M by
where
M = [XCtp, X(t2), ..., X(t )]
2 ni
X(t ) = CxCgtj), x(g tj), ..., x(g Jtj)], j * 1, 2, ..., p
Then C = M A(m)M is the block-diagonal matrix
diafclXt^AOn)), W(t2,A(m)), ..., W(tp,A(m))]
where, for j = 1, 2, ..., p, W(t.,A(m)) is the n. x n. matrix defined by

26 Matrices for Testing Computational Algorithms
w(gt ,A(m))
0
0
w(g -"t^ACm))
w(g t ,A(m)) ...
,-1
0
... w(g J t.,A(m)) 0
To obtain the eigenvalues and eigenvectors, we make use of the fact
that the n x n matrix
0
al
0
0
0
0
a2 '
0
0
0
0
• Vi
a
0
0
0
where a a -a 0 ... a- = 1, has as eigenvalues the n n~th roots of unity
n n*-l n-Z 1 J
and that the eigenvector of D associated with the eigenvalue A is
a-A
n-1
n-2
a2alA
VlV2M'alA
Since the matrices W(t.,A(m)) are all of the same form as D, we can obtain
the eigenvalues and eigenvectors of A(m) from the block^diagonal form C by
applying Theorems 6 and 7.

Construction of Test Matrices 27
As an example, let n = 5, m = 4, k = 3. Then
A(4) =
1
s
2
s
1
0
1
s
2
s
0
1
1
s
0
2
s
1
1
0
s
2
s
1
0
1
s
s
1
27ri
where s = exp —r~ ■ The relation "~" has three equivalence classes, {0,2},
{1}, {3,4}f Hence n.. = n_ = 2 and n„ = 1. Thus the eigenvalues of A(4) are
the square roots of unity (each twice) and unity: 1, 1, 1, -1, -1.
We close this chapter by describing the Vandermonde matrices [34] for
which there is an explicit representation of the inverse. Let
V(Xl,x2,..r,xn) m
^1
2
^2
2
n-1 n-1
Xl X2
n
2
c
n
n-1
n
where the x. are distinct and non-zero. Then if we let V (x-,x ,,
[v. .], we have
v. . = x.b. .
'*„>
where
b = n-i.n-1
k=0 X K
k*i

28 Matrices for Testing Computational Algorithms
and a .. is the sum of all products of m of the numbers x-, x0, .... x. -,
m,n-l l* z* * I-l
x-.i> ••■> x without permutations or repetitions (a- - =1, x-= 0). For
example,
V(l,2,3,4) =
1
2
4
8
1
3
9
27
1
4
16
64
has the inverse
^(1,2,3,4) =
4
-6
4
-1
II
6
2
2
13
3
2
■7 t
■*£ -1
I
6
I
2
2
J.
6
10. Forsythe [78] points out that Varah [79] "also has a program
for generating an arbitrary matrix example, starting from the Jordan form,
and subject to the round-off in inverting the matrix of eigenvectors
(+ principal vectors), and in multiplying matrices. Where he starts with
integers and enough precision, and where the determinant is a small integer,
you can see that there will be no round-off error at all." See examples
5.25, 5.26, and 5.27.

CHAPTER III
TEST MATRICES: INVERSES, SYSTEMS OF LINEAR EQUATIONS,
AND DETERMINANTS
Example 3.
A *
I
~33
-24
-8
If b =
16
-10
-4
+359
-281
- 85
72
-57
-17
, then
X
- A"
A_1 = 1
6
■H-
-1
2
5
-58
48
16
1 •
-16
15
4
-192
153
54
Reference: [29, pp. 120-122].
Example 3,2
A =
~ 1 -2
-2 1
3 -2
1 -1
If b =
3
-4
7
8
>
3
-2
1
5
then x
r
-1
5
3
= A-1b *
A'1
1
1
1
1
1 .
52
•
~-15
-38
-1
-6
-38
-20
-6
16
-1
-6
-7
10
Reference: [4, p. 100].
29

30 Matrices for Testing Computational Algorithms
Example 3.3
A =
1
1+i
1+i
1+2 i
3i
51
2+10i
-5+141
-8+201
A =
10+1 -2+61 -3-21
9-31 81 -3-21
-2+21 -1-21 1
Reference: [60, p. 136],
Example 3.4
A"1-
A =
f2-1
0
0
0
0
.]_
-1
2"2
2"1
0
0
0
0
-1
-1
-2"
2
2
0
0
0
0
1
1
-1
I
■3
•2
•1
0
0
0
1
1
-1
2-4
-2"3
2"2
2"1
0
0 1
0 -1
0 1
0 -1
1 1
1 -1
-2"5
2"4
-2-3
2"2
2"1
,-5
,-3
-1
,-5
Reference: [64],

Inverses, Linear Equations, Determinants 31
Example 3.5
A -
5 7
7 10
6 8
5 7
'—
If b =
|23
32
33
31
Condition numbers:
6
8
10
9
5
7
9
10
—'
, then x !
M(A)
N(A)
P(A)
>
A"1-
r68
-41
-17
10
*—
1
1
1
1
•
= 2720
=
752
- 2984
-41
25
10
-6
-17
10
5
-3
10
-6
-3
2
Reference: [23], [36],
Example 3,6 (See also Example 3.23.)
Let A = [a ,] be the n x n matrix defined by
aij= n
i- j .
,-i
Then A = [b .] is given by
n+2
2h+2 '
if i ■ J ■ 1 or i ■ j ■ n
1 , if i = j and 1 <'i < n
1
\r
2 »
1
if |i - j| - 1 and n t 2
if |i - j| - 1 and n ■ 2
, if |i - j| - n - 1 * 1
2n+2
0 , ifl<|i-j|<n-l

32 Matrices for Testing Computational Algorithms
For example, when n - 4,
A -
3
5
1
2
0
1
10
4
3
2
1
1
2
1
I
~~2
0
3
4
3
2
-
0
J.
1
1
1
2
2
3
4
3
1
2
3
4
1 I
10
0
1
"2
3
"5
Reference: [31], [60, p. 137].
Example 3.7 (Pascal's Matrix)
Let A *= [a .] be the n x n matrix defined by
a^ = a = k £ 0, j « 1, 2, ..., n,
aij = Vl,j + ai,j-l' i,j " 2> 3' •••• n#
Equivalently, we have, for i,j - 1, 2, ..., n,
au k (i-D.'Cj-i): ■
This is called Pascal's matrix because the coefficients of k are obtained
from the Pascal triangle of binomial coefficients.
If k is the reciprocal of an integer, the elements of A are integers,
In addition, det(A) = k
n

For example, if n = 4 and k = — ,
Inverses, Linear Equations, Determinants 33
A =
1
7
1
7
1
7
1
7
1
7
2
7
3
7
4
7
I
7
3
7
6
7
10
7
1
7
4
7
7
20
7
-1_
28
-42
28
-7
det
-42
98
-77
21
(A) =
1
74
28 -7
-77 21
70 -21
-21 7
1_
2401
Reference: [11].
Example 3.8 (The finite segments of the (infinite) Hilbert Matrix-
See example 3-26 for a generalization-)
Let A =
* - m
be the n x n matrix defined by
i+j-1 '
A =
n
I
n
1
2
1
3
1
4
1
n+1
i,j = 1, 2,
I
3
J.
4
1
5
1
n+2
., n.
1
n
1
n+1
1
n+2
2n-l

34 Matrices for Testing Computational Algorithms
If A
-1
n
" M '
then
10
(n) (-l)1+j(n+i-l)!(n+i-l)!
ij (i+j-l)[(i-l)!(j-l)!]2(n-i)!(n-j)!
Alternatively, if t>|- = 1> then for n ^ 1,
(n+l) (n+i)(n+j) (n)
Dij (n+l-i)(n+l-j) Dij '
^-y J *• j £ y ■■■> ^>
b(n+l) = b(n+l)
n+l,j j,n+l
(n+j)[n.'(j-l):]2(n+l-j):
> J - 1» 2, -••> n+l.
Since A is symmetric, we do not exhibit the elements above the main diagonal
in the following matrices:
4
-6 12
9
-36 192
30 -180
180
16
120
240
140
1200
-2700
1680
6480
-4200
2800

Inverses, Linear Equations, Determinants 35
25
-300
1050
1400
630
4800
-18900
26880
-12600
79380
-117600
56700
179200
-88200
44100
36
-630
3360
7560
7560
2772
14700
-88200
211680
-220500
83160
564480
-1411200
1512000
-582120
3628800
-3969000
1552320
4410000
-1746360
698544
7.
49
-1176
8820
29400
48510
38808
12012
37632
-317520
1128960
-1940400
1596672
-504504
2857680
-10584000
18711000
-15717240
5045040
40320000
-72765000
62092800
-20180160
133402500
■115259760
37837800
100590336
-33297264
11099088

36 Matrices for Testing Computational Algorithms
-1
A8 =
64
-2016
20160
-92400
221760
-288288
192192
-51480
84672
-952560
4656960
.-11642400
15567552
-10594584
2882880
2134440000
-2996753760
2118916800
-594594000
11430720
-58212000
149688000
-204324120
141261120
-38918880
4249941696
-3030051024
856215360
304920000
-800415000
1109908800
-776936160
216216000
2175421248
-618377760
176679360
sl-
81
-3240
41580
-249480
810810
-1513512
1621620
-926640
218790
172800
-2494800
15966720
-54054000
103783680
-113513400
65894400
-15752880
38419920
-256132800
891891000
-1748106360
1942340400
-1141620480
275675400
1756339200
-6243237000
12430978560
-13984850880
8302694400
-2021619600
22545022500
-45450765360
51648597000
-30918888000
7581073500
92554285824
■106051785840
63930746880
-15768632880
122367445200
-74205331200
18396738360
45229916160
-11263309200
2815827300

Inverses, Linear Equations, Determinants 37
hi
100
-4950
79200
-600600
2522520
6306300
9609600
8751600
4375800
-923780
326700
-5880600
47567520
-208107900
535134600
-832431600
770140800
-389883780
83140200
112907520
-951350400
4281076800
-11237826600
17758540800
-16635041280
8506555200
-1829084400
8245036800
-37875637800
101001700800
-161602721280
152907955200
-78843164400
17071454400
176752976400
-477233036280
771285715200
-735869534400
382086104400
-83223340200
1301544644400
-2121035716800
2037792556800
-1064382719400
233025352560
3480673996800
-3363975014400
1766086882560
-388375587600
3267861442560
-1723286307600
380449555200
912328045200
■202113826200 44914183600
Condition numbers:
P(A ) = e
x n
M(A ) ~ ke
3.5 n
3.525 n
where k is a constant.

38 Matrices for Testing Computational Algorithms
Determinant:
n
2
3
4
5
6
7
8
9
10
det(A )
n
8.33333
4.62962
65343
74929
36729
83580
73705
72023
16417
33333
96296
91534
51325
98873
26239
01137
43119
92264
33333
29629
39153
15087
58687
26116
91513
24999
31491
33333
62962
43915
16361
73278
93211
01664
86288
86906
33333
96296
34392
32407
88304
98556
20433
94723
05950
( -2)
( -4)
( -7)
(-12)
(-18)
(-25)
(-33)
(-43)
(-53)
Reference: [20], [43], [50], [55].
Example 3.9
Let A = [a. .] be the n x n matrix given by
n ij ° J
•ij " X>
j = 1, 2, ..
ai:j - (i+j-1) , i = 2, 3, ..
A =
n
1
2
1
3
I
n
1
3
1
4
j.
4
1
5
1 1
n+1 n+2
n.
n, j=l, 2, ..., n.
1
n+1
1
n+2
2n-l
If A-1 = [b. .], then
n ijJ'
bu = (-i)
1 - 1, 2,
\j+l ' <
for i = 1, 2,
n, j = 1, 2, ..., n-1.

Inverses, Linear Equations, Determinants 39
Furthermore,
det(An) = (-D^V1,
where
8n+l " („-?) ('") »■*»«>. — Sl - l-
The condition numbers of A are given by
P(A ) « C, 25n log n
n i
M(A ) = C, n 25n
n Z
-3 -3
where Cn = 8 x 10 and C0 = 4 x 10 . The condition numbers of A ,
12 n*
n = 2, 3, ..., 10, rounded to 5 significant digits, are given in the following
table:
n
2
3
4
5
6
7
8
9
10
M(An]
12
540
17280
67200
23814
80681
28333
95447
33640
X
X
X
X
X
X
1
101
io3
io4
io6
io7
109
P(An)
12.587
354.51
13090
45057 x 101
15259 x IO3
51270 x IO4
17164 x 106
57364 x 107
19158 x IO9
When n = 6, the inverse is given by

40 Matrices for Testing Computational Algorithms
a;1 =
6
-6
105
1 -560
1260
-1260
462
Reference: [33].
Example 3
.10
630
-7350
29400
-52920
44100
-13860
-6720
88200
-376320
705600
-604800
194040
22680
-317520
1411200
-2721600
2381400
-776160
-30240
441000
-2016000
3969000
-3528000
1164240
13860
-207900
970200
-1940400
1746360
-582120
Let A = [a ] be the n x n matrix defined by
a. .=
if i < j,
if i > j.
If k~l = [b^], then
/
4i"
2 '
41-1
n
2n-l
bii- <
A =
if i = j and i < n
if i = j = n
e,
1
1
2
1
3
V
when
1
2
1
2
3
, 1(1+1)
2i+l '
1(1+1)
2j+l »
o ,
n = 3,
1 "
3
2
3
1
if j =
i + 1
if 1 = j + 1
if |i-j| >1.
A"1-
F4 2
3 '3
2 32
"3 15
o -1
0
6
" 5
9
5
Reference: [43], [60, pp. 138-139].

Inverses, Linear Equations, Determinants 41
Example 3.IX
Let A = [a ] be the n x n matrix defined by
1J
-(£)*"$5)
Then A is orthogonal, and A = A.
Reference: [43],
Example 3.12
Let A = [a .] be the n x n matrix defined by
lu
a.± = n + 1 - i, if i > j.
A =
n
n-1
n-2
2
1
n-1
n-1
n-2
2
1
n-2
n-2
n-2
2
1
2
2
2
2
1
1
1
1
1
1
A"1-
1 -1
■1 2
-1
•1
2
2
■1
■1
2
n x n.
Reference: [24].

42 Matrices for Testing Computational Algorithms
Example 3.13
Let A = [a. ] be the n x n matrix defined by
A =
1
2
3
n-1
n
-1
1
A"1-
a± = max(i,j), i,j * 1, 2, ..., n.
2
2
3
n-1
n
3
3
3
n-1
n
n-1
n-1
n-1
n-1
n
1
■2
1
1
-2
■2
1
n-1
n
, n x n.
Reference: [5],
Example 3.14 (Hadamard Matrices)
define
Let p be a prime, p _> 3, and let n » p - 1. For an arbitrary integer k,
0, if p divides k,
I ™" 1 " \ 1> if k is congruent to a square mod p,
■1, otherwise.
Define A * [a. .] to be the n x n matrix such that
•U
ft)-
i,j - 1, 2, ..., n.

Inverses, Linear Equations, Determinants 43
If A-1 = [bj.], then
'11-*&?)-(*)-(*)]•
i,j - 1, 2,
n.
For example, when p = 5,
A =
■1
-1
1
0
■1
1
0
1
1
0
1
0
1
■ 1
J.
5
•3
■1
1
■2
-1 1
3 2
2 3
1 -1
Condition number:
Reference: [43].
P(A) = /p
Example 3.15
Let B be the n x n matrix (row elements are binomial coefficients except
for sign)
B =
0
-1
-2
-3
-4
0 0
0 0
1 0
3 -1
6 -4
0
0
0
0
1
-1 T
Then B = B. Furthermore, if A » B B, then a
ij
■ (7)
, i,j = 0,1,2,...,n-l,
-1 T
and A = BB . The eigenvalues of B are all of modulus 1, and the eigenvalues
of A occur in reciprocal pairs.
Condition number:
P(A) = exp (4n log 2)
Reference: [42, pp. 240-241], [60, p. 140].

44 Matrices for Testing Computational Algorithms
Example 3.16
A =
n
1 2
n-l
n-l
1
2
n-l
n
If A = [b ] and k = , .1wo—TT »
n ij n(n+l)(2n-5)
b±± = 1 - ki , if 1 < i < n-l
b = -k
nn
bi. = -kij, if i t j, 1 < i < n-l, and 1 < j < n-l
b, = b . = ki, if 1 < i < n-l.
in nx ' — —
Determinant:
det(A )
n
I
k
Condition numbers (rounded):
n
2
3
4
5
6
7
8
9
0
M(An)
8
9
14.4
24
35.2653
48.4167
63.5152
80.5846
99.6364
P(An)
6.854102
9.89898
6.531124
8.830950
11.32624
14
16.83897
19.83240
22.97141
Reference: [l].

Example 3.17
Inverses, Linear Equations, Determinants 45
where
A =
1
a 1
1 a
1 a
1
1
a
n x n.
A =r— [a..] is the n x n matrix defined by
b ij J
n J
/
b. .b ,,
l-l n-i'
>+Jv
LaJi»
if i = j
(-l)"'Jb, .b ., if j > i
i-1 n-j' J
if j < i
bo = 1
b]L-a
bk = abk-l " bk-2» k ■ 2« 3>
n.
Reference: [10],
Example 3.18
A =
2
■1
■1
2
■1
■1
2 -1
n x n.

46 Matrices for Testing Computational Algorithms
A = —— C where C = [c. .] is the n x n matrix defined by
n+1 ij J
*U
i(n-
|Ci,J
lCJi'
i+D,
-1 '
i.
if
if
if
i
j
j
=
>
<
j
i
i
For example, when n = 4,
A =
2 -1
■1 2
-1
-1
2 -1
-1 2
A 5
3
6
4
2
2 1
4 2
6 3
3 4
Condition number:
Reference: [31], [43]
Example 3.19
P(A) ~
4n<
V
Let n be an odd integer, and let A . be the n+1 x n+1 matrix
n+1
0 x,
yx o x2
y2 0 x3
Vl ° *n
*n °
with x # 0 and y ± 0, i = 1, 2, ..., n.

Inverses, Linear Equations, Determinants 47
a;1-
For n J> 3,
n+1
0
a
m
0
0
n-1
0
a
1
• ■ •
• • •
0
a2
0
0
! °
i °
; o
1 •
! °
i 0
i °
i ai
b
0
b
b
0
b
0
m
m-1
n+1
where m = —r— and
and
/
V.
n
, lxk+l k-1 y
Xn j=l Xn-2j
/
bk =
V
(-l)k+1 JT1 V21+1
yn j=l yn-2j
if k = 1
if k = 2, 3,
if k = 1
if k - 2, 3,
m.
m.

48 Matrices for Testing Computational Algorithms
Determinant:
det(An+1) = (-1)
(n+l)/2
n+1
.2
Reference: [12].
Example 3.20
A =
(x+b) 1
1 x
1
1
x
lxl
1 (x+a)
n x n.
Inverse:
where
[c. .1 is the n x n matrix defined by
(-Di+jr. ,s .
1-1 n-i .- . . .
c. . = c.. = ' , N ■ , if j < l,
ij ji (x+a)r -- r ' J -
J J N ' n-1 n-2
and
ro =
rl =
rk =
so =
sl =
1
x + b
xrk-l *
1
x + a
• rk-2
rV-2* K-2, 3, ••-, n-1,
sk ~ XSk-l " Sk-2> ~ * * *'*> n"^'

Inverses, Linear Equations, Determinants 49
For example, if
A =
■3
1
is an n x n matrix, then
I
2
1
■2 1
1 -2
1
3
Reference: [18, pp. 33-36], [43],
1
3
5
1 -2
1
1
3
5
2n-l
Example 3.21
Solution:
(1-10'
1
1
1
n)
-1
-1
-1
0
1
1
0
0
= 10
n
x„ =
x„ =
X, =
10n + 1
10n + 2
10n + 3
-l I rXlI [-3
-1 x2 -2
0 x3 -1
-1 x, -3
114 1 I
Reference: [52],

50 Matrices for Testing Computational Algorithms
Example 3.22
det
det
73
92
80
73
92
80
78
66
37
78
66
37
24
25
10
= 1
24
25
10.01
= -118.94
det
-73 78 24
92.01 66 25
-80 37 10
= 2.08
det
-73
92
-80
78.01
66
37
24
25
10
= -28.20
Reference: [27], [56].

Inverses, Linear Equations, Determinants 51
Example 3.23 (see also Example 3.6)
Let A = [a. .] be the n x n matrix defined by
atj = |i-JI
Then
^n^
-1
1
n-1
-1
-1 2
-1
1
n-1
^"n^
det A = (-l)n_12n"2(n-l).
An interesting generalization of this matrix is also known [77, p. 32]
Reference: [77, p. 31]

52 Matrices for Testing Computational Algorithms
Example 3.24
For arbitrary constants a., a„, ..., a ., let A
.... a ,) = [a..] denote the n x n matrix defined by
' n-1' ijJ J
r
1,
■u- <
y
j £ i,
j < i-
For example,
A4 =
1111
ax 1 1 1
al a2 l l
al a2 a3 l
If we let A = [b. .], then, for n > 1,
n lj
r
1/Cl-a^,
ij
i = j, i ^ n,
-l/Cl-a^, j f 1+1, i ±
J (aj_1-aj)/[(l-aj)(l-aj_1)], i = n, j ^ 1,
-a^/il-a.^), i = n, j = 1,
i = j = n,
0, otherwise.
W-Vi)'
For n = 1, A± = [1] .
The determinant of A is given by
n ° J
det(An) = (l-ai)(l-a2) ... d-an-1).
Hence A is singular if and only if a. =1 for some i.
Reference: [75]

Inverses, Linear Equations, Determinants
Example 3.25 (Combinatorial Matrix)
C = [y*61;Jx] =
(x+y)
y
y
y
(x+y)
y
y
y
(x+y)
y
y
y
(x+y)
det (C) = x " (x+ny) ,
If C" = [b. .] , then £ b. . = —-"- , and
ij , , iJ x+ny
6.. (x+ny)-y
b = —=J
ij x(x+ny)
For example, if n = 3, x=2, y = 1, then
C =
3 11
13 1
113
_1
10
4 -1 -1
-1 4 -1
-1 -1 4
and
det (C) = 20,
. L. bij = 5 '
Reference: [81, p. 36].

54 Matrices for Testing Computational Algorithms
Example 3-26 (Cauchy's Matrix)
A =
. x.+x.
L 1 J
(xi+yi)_ (Ki+y2)~
(x2+y1)" (x2+y2)~
• <xi+yn>
(x +y- ) (x +y^)
(x2+yn)
-1
(x +y )
-1
det(A) =
TT (x.-x^Cy .~y±)
lgKjgn
TF <vyi)
If A = [b^] , then
b.. =
TT (xi+yk)<xk+yi)
lgkSn
(x.+y.)
'TT (x-x^)"
lgkgn J K
lk*j J
^TT (yi-y/l
l^kgn X K
k^i J
and Eb = (Xj+x +. . .+Xn) + (y +y2+... +yn) .
Note: The finite segments of the (infinite) Hilbert matrix are special
examples of Cauchy!s matrix. See example 3.8.
Referencel [81, p. 36].
Example 3.27 (Vandermonde!s Matrix)
See Section 9 of Chapter II.

CHAPTER IV
TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF REAL SYMMETRIC MATRICES
Example 4,1
5
4
1
1
4 1
5 1
1 4
1 2
1
1
2
4
Eigenvalues:
A, =
A. =
A. =
A, =
10
5
2
1
Eigenvectors:
xl =
X- =
■1
■1
2
2
x„ =
0
0
-1
1
x, «
■1
1
0
0
Reference: [49, pp. 54-55]
55

56 Matrices for Testing Computational Algorithms
Example 4.2
4
6
1
4
4 1
1 4
6 4
4 6
Eigenvalues:
Eigenvectors:
xi=
Ax = 15
A2 = 5
A3 - 5
\k--i
|l
1
1
1
9
x„ =
-1
■1
1
1
x, =
1
-1
-1
1
Note: For A„ = A_, we have a two-dimensional subspace of eigenvectors
corresponding to this multiple eigenvalue. x_ and x„ are two orthogonal vectors
from this subspace.
Reference: [49, pp. 53-54], [60, pp. 145-146],
Example 4.3
2
1
3
4
1
-3
1
5
3 4
1 5
6 -2
-2 -1

Eigensystems—Real Symmetric Matrices
Eigenvalues:
X2*
-8.0285 7835
7.9329 0471
5.6688 6437
-1.5731 9073
Eigenvectors:
xl *
X3 =
1.0000 0000
2.5014 6029
-0.7577 3064
-2.5642 1169
f1.0000 0000
0.9570 0150
-1.4204 6822
1.7433 1690
X2 *
• 1
x4 =
1.0000 0000
0.3778 1815
1.3866 2122
|0.3488 0573
1.0000 0000
-0.9070 9211
-0.3775 9122
-0.3833 3124
Reference: [3], [8], [49, pp. 66-67], [74],
Example 4.4
5
7
6
5
7
10
8
7
6
8
10
9
5
7
9
10
Characteristic polynomial:
P(A) = 1 - 100A + 146A2 - 35A3 + A4
Eigenvalues:
A i 30.28868
A = 3.85806
A3 = 0.84311
A. = 0.01015
4
Inverse: Example 3.5
Reference: [27], [42, pp. 247-248], [49, p. 53],

58 Matrices for Testing Computational Algorithms
Example 4.5
0.81321
-0.00013
0.00014
0.00011
0.00021
-0.00013
0.93125
0.23567
0.41235
0.41632
0.00014
0.23567
0.18765
0.50632
0.30697
Tridiagonal form from Householder's method:
0.00011
0.41235
0.50632
0,27605
0.46322
0.00021
0.41632
0.30697
0.46322
0.41931
a.
0.81321
0.57378
1.33978
0.06519
-0.16450
0.00030
■0.48980
■0.44013
0.17294
Eigenvalues:
A, =
K *
A, =
A, =
A, =
1.67828
0.81321
0.41985
0.01521
-0.29908
Reference: [67],
Example 4.6
5
4
3
2
1
4 3
6 0
0 7
4 6
3 5
2
4
6
8
7
1
3
5
7
9

Eigensystems—Real Symmetric Matrices 59
Eigenvalues:
Ax = 22.4068 7532
A2 = 7.5137 24155
A3 = 4.8489 50120
A. = 1.3270 45605
4
An = -1.0965 95181
Reference: [58], [68], [77]
Example 4.7
10
1
2
3
4
1
9
-1
2
-3
2
-1
7
3
-5
3
2
3
12
-1
4
-3
-5
-1
15
Tridiagonal form from Householder's method:
a.
x
9.295202 17754
11.626711 5560
10.%0439 2078
6.11764.7 05885
15.000000 0000
0.749484 677741
-4.496268 20120
•2.157040 99085
7.141428 42854
Eigenvalues:
Ax = 1.655266 20775
A2 = 6.994837 83064
A3 = 9.363554 92016
A4 i 15.808920 7645
K * 19.175420 2773
Reference: [37],

60 Matrices for Testing Computational Algorithms
Example 4.8
5
1
2
0
2
5
1
6
-3
2
0
6
-2
-3
8
-5
-6
0
0
2
-5
5
1
-2
-2
0
-6
1
6
-3
Eigenvalues:
Reference: [37],
Ax = -1.598734 29358
A2 i -1.598734 29346
A3 = 4.455989 63847
A4 = 4.455989 63855
A5 i 16.142744 6551
A, = 16.142744 6553
6
Example 4.9
Eigenvalues:
1
2
3
0
1
2
Ax « 12
A2 = 12
A3
= 0
2
4
5
-1
0
3
.4113
.4113
.2849
3
5
6
-2
-3
0
3643
3642
0
-1
-2
1
2
3
864395
1
0
-3
2
4
5
A. = 0.2849 864365
4
A5 = -1.6963 22849
A6 =5= -1.6963 22851
Reference: [58], [68], [77].

Eigensystems—Real Symmetric Matrices 61
Example 4.10 (Rosser, et al.)
A =
611
196
192
407
-8
-52
-49
29
196
899
113
-192
-71
-43
-8
-44
-192
113
899
196
61
49
8
52
407
-192
196
611
8
44
59
-23
-8
-71
61
8
411
-599
208
208
-52
-43
49
44
-599
411
208
208
-49
-8
8
59
208
208
99
-911
29
-44
52
-23
208
208
-911
99
Tridiagonal form from Lanczos' Method for 10 A:
a*
i
0.899
0.1086629633
0.7859177671
■0.7935214279
0.003963315517
1.0160663075
1.0199110708
1.0000000030
0.096939
0.039517948848
0.4088977136
0.0520498144977
0.004021099703
0.1070421101 x 10"
0.7048359779 x 10*
8
10
Eigenvalues:
*x - 10 (/10405 = 1020.04901843
A = 1020
A3 - 510 + 100 \/26 = 1019.90195136
\ = 1000
4
A5 = 1000
A, = 510 - 100 \/26 = 0.09804864072
o
A? = 0
AQ = -10 V10405 = -1020,04901843

62 Matrices for Testing Computational Algorithms
Eigenvectors:
2
1
1 ■
2
102 - /10405
102 - \/l0405
Xl =
■204 + 2^10405
■204 + 2\/l0405
X5 =
7
14
■14
-7
-2
-2
-1
-1
x, =
x„ =
, x3
2
-1
1
-2
5 - \/26
-5 -H\/26
-10 + 2y/2?
10 - 2^/26
2
-1
1
-2
5 +\/26
-5 - \f26
10 - 2^2?
10 + 2\/26
' X4 =
1 J
-2
-2
1
-2
2
-1
1
• X7
1
2
-2
-1
14
14
7
7
» X8
102 + \A0405
102 + \/1040 5
•204 - #10405
■204 - 2k/l0405
Note: For A, = A,«e have a two-dimensional subspace of eigenvectors
corresponding to this multiple eigenvalue, x, and x are two orthogonal vectors
from this subspace.
Reference: [47],

Eigensystems—Real Symmetric Matrices 63
Example 4.11
5
2
1
1
2
6
3
1
1
1
3
6
3
1
1
1
1
3
6
3
1
1
1
1
3
6
3
1
1
1
1
3
6
3
1
1
Eigenvalues:
A, =
A„ =
A„ =
A, =
A. =
A, =
A, =
A„ =
A„ =
10
11
14.94181
12.19615
8.82842
6.00000
4.4G664
4.12924
4.00000
4.00000
3.17157
1.80384
0.52228
1
1
3
6
3
1
1
93276 76382
24227 06632
71247 461900
00000 000000
99006 731521
84841 890931
00000 000000
00000 000000
28752 538100
75772 933680
22874 6137256
1
1
3
6
3
1
1
1
1
3
6
3
1
1
1
3
6
2
1
1
2
5
Reference: [49, pp. 78-79], [66].

64 Matrices for Testing Computational Algorithms
Example 4.12
0.25000
0.06675
0.04000
0.02475
0.07050
0.06375
0.06925
0.02050
0.03600
-0.01025
-0.00175
0.02750
0.02300
0.00200
0.06675
0.25000
0.10400
0.07475
0.03625
0.11675
0.11050
0.06225
0.05100
0.03250
0.02400
0.03600
0.06350
0.05300
0.04000
0.10400
0.25000
0.14575
0.03725
0.07175
0.07800
0.12200
0.11275
0.09375
0.10175
0.09600
0.14300
0.11550
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.02050
0.06225
0.12200
0.12800
0.05700
0.08800
0.08725
•■'0.25000
0.14100
0.13275
0.15550
0.13050
0.11825
0.09125
0.03600
0.05100
0.11275
0.12475
0.05050
0.07150
0.06950
0.14100
0.25000
0.07425
0.10750
0.09175
0.10725
0.08225
-0.01025
0.03250
0.09375
0.10550
0.01475
0.04850
0.03725
0.13275
0.07425
0.25000
0.15500
0.09625
0.09950
0.09425
-0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
Tridiagonal form from Givens f Method:
0
0
0
0
0
0
0.
0.
0.
0.
0.
0.
0.
0.
.25000
.76849
.91955
.23093
.13305
22254
11612
12033
12371
12856
10776
13703
13805
10379
0000
1173
6756
8895
3788
9575
7856
9373
9912
1407
8089
9203
7030
6943
02475
07475
14575
25000
05375
07000
05225
12800
12475
10550
13000
14575
13975
13375
0.07050
0.03625
0.03725
0.05375
0.25000
0.04575
0.05750
0.05700
0.05050
0.01475
0.04500
0.07150
0.05300
0.01600
0.06375
0.1167 5
0.07175
0.07000
0.04575
0.25000
0.08625
0.08800
0.07150
0.04850
0.03200
0.04475
0.03300
0.04500
0.06925
0.11050
0.07800
0.05225
0.05750
0.08625
0.25000
0.08725
0.06950
0.03725
0.04025
0.04300
0.04075
0.01450
00175
02400
10175
13000
04500
03200
04025
15550
10750
15500
25000
13350
14850
13050
0.02750
0.03600
0.09600
0.14575
0.07150
0.04475
0.04300
0.13050
0.09175
0.09625
0.13350
0.25000
0.11100
0.10075
0.02300
0.06350
0.14300
0.13975
0.05300
0.03300
0.04075
0.11825
0.10725
0.09950
0.14850
0.11100
0.25000
0.14325
0.00200
0.05300
0.11550
0.13375
0.01600
0.04500
0.01450
0.09125
0.08225
0.09425
0.13050
0.10075
0.14325
0.25000
h
0.15366 0746
0.46726 0328
0.11925 6498
0.08076 3539
0.03394 7196
0.03609 0904
0.03502 2375
0.02915 7561
0.03745 3705
0.01609 0599
0.02382 6467
0.02946 8449
0.00764 6394

Eigensystems—Real Symmetric Matrices 65
Eigenvalues:
\
\
h
\
s
\
s
*8
N.
\o
\l
\2
A13
\*
=
s
=
=
=
=
=
=
=
=
=
=
=
=
1.33403
0.46276
0.26773
0.23163
0.17735
0.17130
0.16632
0.14342
0.12278
0.10321
0.09720
0.08422
0.07359
0.06437
48369
62026
32979
94839
63338
75618
46020
28761
75231
565070
942646
540935
784262
214251
005174
889680
463870
815480
51624 309478
92161
52705
71188
99133
618017
646369
537465
667302
Eigenvectors:
We give the first four eigenvectors of the tri-diagonal matrix because the later
components of these vectors are very small. The other eigenvectors are quite normal
in form.
0.12180621
0.85930971
1.00000000
0.10864611
0.00731255
0.00022356
0.00000663
0.00000019
0.00000001
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.72220468
1.00000000
-0.89179298
-0.50225957
-0.12487537
-0.01793071
-0.00188653
-0.00019439
-0.00001692
-0.00000190
-0.00000009
-0.00000000
0.00000000
0.00000000
0.49474263
0.05709601
-0.22388762
1.00000000
0.78617724
0.73992634
0.18686459
0.04640116
0.01011081
0.00275399
0.00028519
0.00005483
0.00001250
0.00000059
-0.34701603
0.04146403
0.06647838
-0.54593352
-0.10289858
1.00000000
0.34864816
0.11940974
0.03703434
0.01375120
0.00188770
0.00052743
0.00016690
0.00000998
Reference: [9], [63], [66], [67].

66 Matrices for Testing Computational Algorithms
Example 4.13 (Hilbert Matrix)
Let A = |a^V'| be the n x n matrix defined by
s> - [#]
tf
i+j-1 '
i,j - 1, 2,
n.
A =
n
n
I
2
1
3
I
4
1
n+1
1
3
1
4
J.
5
1
n+2
n
1
n+1
1
n+2
2n-l
Eigenvalues and Eigenvectors:
The eigenvalues and eigenvectors of A , n = 3, 4, ..., 10, are given on
the following pages. In addition, we give the eigenvalue of largest magnitude
and
the corresponding eigenvector for A„ and A__.
Eigenvalue
1.26759 188
Eigenvector
1.00000 000
0.53518 376
Inverse: Example 3.8
Reference: [19], [20], [77, p. 30]
"20
Eigenvalue Eigenvector
1.90713 472
1.00000 000
0.63153 893
0.48170 552
0.39577 939
0.33864 052
0.29732 839
0.26579 806
0.24080 108
0.22041 627
0.20342 569
0.18901 536
0.17661 823
0.16582 577
0.15633 540
0.14791 772
0.14039 536
0.13362 876
0.12750 652
0.12193 851
0.11685 095

ORDER OF MATRIX = 3
ElGENVALUtS
1.40831 89271 23o54(-00>
EIGENVECTOR!
1.00000 00000 OOOOO(-OO)
5.56Q32 55563 05693(-01)
3.90907 94792 51080(-01)
EIGENVALUES
1.223*7 065§5 39058(-01>
EIGENVECTORS
*8.4|517 43276 29785(-01)
8.13998 17376 62614(-01)
1.00000 00000 OQGUO(-OO)
2.68734 03557 73d29(-03>
-1.78857 98^3| 23438(-0U
1.00000 00000 OOOOO(-OO)
-9.64868 00204 5$515(-01)
ElGENVALUtS
1.50021 42b00 59*43(-00)
ORDER OF MATRIX
EIGENVECTORS
1.00000 0Q000 OOOOO(-OO)
5.70172 08366 32358(-01)
4#06778 98£Q2 75292(*01)
3.18146 96887 37940(-01)
EIGENVALUES
1.69141 22022 14500(-01)
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
-6.36518 90190 07507(-01)
-8.75450 79607 67703(-01>
-8.83129 58721 03381(-01)
6.73827 ^6057 60/48(-03)
*2.41517 71638 JS848(-01)
1.00000 OOQOO OGOOG(-OO)
-1.35093 31925 07#54(-01)
-8.60314 35862 04442(-01)
9.67023 04022 58689(-05)
3.68876 82614 l4l05(-02)
-4.1S349 28778 03112<-01>
1.00000 00000 OOOOO(-OO)
-6.50171 21973 36798(-01)
ElGENVALUtS
1.56705 06^10 98^31 (-00)
ORDER OF MATRIX * 5
EIGENVECTORS
EIGENVALUES
1.00000 00000 OOOOO(-OO)
5.80566 92249 80478(-01)
4.18800 95256 90560(-01)
3.30061 05409 17674(-01)
2.73258 24401 62320(-01)
2.08534 21861 10133(-01)
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
-4.58425 80576 61740(-01)
-7.05925 82907 15063(-01)
•7.37537 92074 31147(-01)
•7.12798 94314 80946(-01)

tlGtNVALUtS
1.14074 91623 4196K-02)
ORDER OF MATRIX
ElGtNVECTORS
•2.95833 43954 91379(-01)
1.00000 00000 OOOOO(-OO)
1.66348 46563 67509(-01)
-4.27528 04665 91248(-01)
-7.80543 77407 62442(-01)
3.28792 *tldl nob3(-06>
-8.04735 96573 69526(-03)
1.52103 86654 527l8(-01)
-6.59762 08136 21921(-01)
1.00000 00000 OOOOO(-OO)
-4.90419 53143 50719(-01)
EIGENVALUES
1.61889 98589 24339C-00)
8
ORDER OF MATRi;
ElGtNVECTORS
1.00000 00000 OOOOO(-OO)
5.88628 54342 55432(-01)
4.28327 28442 89561(-01)
3.39661 89183 87095<-01)
2.82523 58794 21492(-01)
2.42337 81112 28495(-01)
1.63215 21J19 87D82(-02)
-3.44477 74040 00321(-01)
1.00000 00000 OOOOO(-OO)
3.31669 05639 78445(-01)
-1.90443 48397 72404(-01)
-5.19908 55937 27446(-01)
-7.20650 57788 73129(-01)
1.25707 57122 62D19C-05)
1.84443 82298 42188(-02)
-2.97466 27961 49800(-01)
1.00000 00000 OOOOO(-OO)
-7.34137 29699 37382(-01)
-7.30764 18529 36246(-01)
7.59856 90405 64665(-01)
5 (CONT.)
EIGENVALUES
3.05898 04015 11917(-04)
* 6
EIGENVALUES
2.42360 87057 52096(-01)
6.15748 35418 26577(-04)
1.08279 94845 65550C-07)
EIGENVECTORS
7.06702 26210 87525(-02)
-6.48336 02593 66261(-01)
1.00000 00000 OOOOO(-OO)
3.49178 63233 06241(-01)
-8.35542 93387 42830(-01)
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
-3.43477 76103 67806(-01)
-5.95389 61269 85598(-01>
-6.42274 99431 02546(-01)
-6.31671 46805 61395(-01)
-6.03204 01490 85321(-01)
-1.15089 16226 58221(-01)
9.07815 69634 66591(-01)
-9.90373 92462 04362(-01)
-7.71318 49997 79162(-01)
8.69902 39991 00457(-02)
1.00000 00000 OOOOO(-OO)
1.80948 25414 40515(-03)
-5.16182 53594 24858(-02)
3.48907 75253 55039(-01)
-9.06717 68457 84127<-01)
1.00000 00000 OOOOO(-OO)
-3.93741 11149 37020(-01)

EifcENVALUtS
1.66088 53389 26*31(-00)
2.12897 b4908 32/95(-02)
2.93863 68X45 92969<-0b>
3.49389 db0b9 91<!l8(-09)
ORDER
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
5.9bl22 63106 51334(-01)
4.36126 49735 70395(-01)
3.47622 28057 85343(-01)
2.90284 56422 45982(-01)
2.49777 30606 63215(-01)
2.19495 43192 32110C-01)
-3.88993 12692 28422(-01)
1.00000 00000 OOOOO(-OO)
4.40423 18707 70565(-01)
-3.43693 09768 67312(-02)
-3.48496 99421 85792(-01)
-5.48611 13060 83639(-01)
-6.74584 79000 81032(-01)
2.54375 48028 50871f-02)
-3.62466 87695 55796(-01)
1.00000 00000 OOOOO(-OO)
-3.18671 25647 34205(-01)
-7.90476 81764 75158(-01)
-2.94027 65031 95887(-01)
7.64617 63467 28869(-01)
3.59098 91821 95847(-04)
-1.44149 72273 50558(-02)
1.39474 73803 77168C-01)
-5.44035 08875 84887(-01)
1.00000 00000 OOOOO(-OO)
-8.65947 69018 12042(-01)
2.84831 36565 59360(-01)
7
EIGENVALUES
2.71920 19814 93452(-0l)
1.00858 76107 70142(-03>
4.85676 33615 74250(-07)
EIGENVECTORS
l.OOOQO 00000 OOOOO(-OO)
-2.61651 87231 55985(-01)
-5.15876 57109 07207(-01)
-5.73403 92060 79247(-01)
-5.72924 4M)712 20064(-01)
-5.52870 16536 01293(-01)
-5.26499 39668 34787(-01)
-1.42730 60664 90882(-01)
1.00000 00000 OOOOO(-OO)
-8.08037 82159 87594(-01)
-8.76415 71705 77919(-01)
-3.24986 39912 94217(-01)
3.46758 27944 15234(-01)
9.67685 27364 04640(-01)
-3.82934 35999 28926i-03)
9.58555 22029 61430(-02)
-5.40843 92603 82367(-01)
1.00000 00000 OOOOO(-OO)
-2.70501 09557 80551(-01)
-8.43244 13447 80657(-01)
5.65766 58320 00075(-01)

ORDER OF MATRIX * 8
EIGENVALULS
U69593 89969 21**9 (-00)
2.62128 4Jb78 11905<-02>
5.43694 J3697 49942(-05)
1.79887 374b8 l/s77<-08>
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
6.00504 2457b 79538(-01)
4.42671 55401 19186(-01)
3.54370 44699 96978(-01)
2.96918 57844 45071(-01)
2.56180 92948 69805(-01)
2.25629 36880 82276(-01)
2.01790 18703 79183(-01)
-4.30353 53605 31482(-01)
1.00000 00000 OOOOO(-OO)
5.19670 85157 87076(-01)
7.95547 21960 88871(-02)
-2.23317 97982 52556(-01)
-4.23090 40968 60939(-01)
-5.53618 29278 79063(-01)
-6.38180 47813 30543(r01)
3.30983 85076 81348(-02)
-4.26438 76153 95457(-01)
1.00000 00000 OOOOO(-OO)
-5.12022 50480 36401(-02)
-6.72019 76374 56363(-01)
-5.85721 70581 51348(-01)
-2.88541 78933 63593(-fc2)
7.65890 29707 01414(-01)
-9.20944 89718 87311(-04)
3.30647 67571 34515(-02)
-2.77763 32839 45767(-01)
8.69385 31714 42472(-01)
-9.95882 21590 46623(-01)
-1.31307 33940 75290(-01)
1.00000 00000 OOOOO(-OO)
-4.97329 08134 30048(-01)
EIGENVALUES
2.98125 21131 69307<-«U>
1.46768 81177 4l867(-03)
1.29433 20918 7281H-06)
1.11153 89663 72442(-10)
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
-1.99641 07668 62729(-01)
-4.55006 08109 55082(-01)
-5.20378 81437 80070(-01)
-5.27565 95376 14758(-01)
-5.13976 52248 24774(-01)
-4.92889 11462 14828(-01)
-4.6961.7 42309 87346(-01)
-1.57264 17782 81082(-01)
1.00000 00000 OOOOO(-OO)
-6.11217 36807 18912(-01)
-8.34786 09555 82616(-01)
-5.01055 24897 20307(-01)
-2.04888 90803 70457(-02)
4.52548 32316 57058(-01)
8.67560 91295 41801(-01)
-6.57331 00511 27815(-03)
1.47989 76372 18060(-01)
-7.22416 37481 59994(-01)
1.00000 00000 OOOOO(-OO)
2.12540 06545 33205(-01)
-7.21939 02421 98936(-01)
-6.07829 90639 21476(-01)
7.04255 03469 70941(-01)
-6.86103 92145 12811(-05)
3.68787 70518 27661(-03)
-4.82672 54524 49843(-02)
2.61713 39967 6104K-01)
-7.05747 34717 96188(-01)
1.00000 00000 OOOOO(-OO)
-7.12509 13818 01248(-01)
2.01241 83438 37764(-01)

OHOEK OF MATRIX s 9
tibt'^VALULS
1.7258b 26609 0164M-00)
3.10389 2bf81 26633(-02)
8.756d8 50514 b9/57(-0b)
5.38561 3348b 22494C-08)
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
6.05062 73643 51117(-0l)
4.48271 58106 99431(-01)
3.60192 03013 69706(-01)
3.02681 26027 11120(-01i
2.61776 14674 05719(-01)
2.31016 02171 39396C-01)
2.06956 70822 18468(-01)
1.87577 43844 56729(-01)
-4.69215 60414 24556(-01)
1.00000 00000 OOOOO(-OO)
5.81298 48942 56370(-01)
1.68338 31445 98111(-01)
-1.25629 34840 26744(-01)
-3.25116 29771 00839C-01)
-4.59281 69753 28175(-01)
-5.49127 85754 13251(-01)
-6.08710 62023 19202(-01)
4.13830 29491 86680(-02)
-4.90024 66072 79400C-01)
1.00000 00000 OOOOO(-OO)
1.41406 98631 98621(-01)
-5.29978 89637 04739(-01)
-6.54975 57014 80830(-01)
-3.71913 95311 21723(-01)
1.44275 27836 59178(-01)
7.66821 37097 76299(-01)
-1.57026 36778 04265C-03)
5.13643 34130 24l27(-02)
-3.63966 02990 96905C-01)
1.00000 00000 OOOOO(-OO)
-7.07732 14392 1262K-01)
-6.33233 77263 71795C-01)
4.61853 32026 77920(-0l)
8.23643 49129 50087C-01)
-6.11750 86298 36100C-01)
EIGENVALUES
3.21633 12229 92068(-01)
1.97893 38602 15924(-03)
2.67301 34105 99414C-06)
6.46090 54226 38582(-10)
EIGENVECTORS
1.00000 00000 OOOOO(-OO)
-1.50563 00038 85823C-01)
-4.06370 51543 62231(-01)
-4.77760 59330 52471(-01)
-4.90983 21999 98b90(-01)
-4.82553 42744 47961(-01)
-4.65723 08579 25155(-01)
-4.45945 07919 47498(-01)
-4.25620 54046 16327(-01)
-1.70653 85569 90212(-01)
1.00000 00000 OOOOO(-OO)
-4.64717 14467 22524(-01)
-7.75951 09562 54300(-01)
-5.78327 97815 76324(-01)
-2.21347 44112 47116(-01)
1.53939 04201 98640(-01)
4.96040 77413 22528(-01)
7.89930 71224 23032(-01)
-1.05472 67848 64242(-02)
2.17325 36475 07308(-01)
-9.40220 91065 07029(-01)
1.00000 00000 OOUOO(-OO)
5.70068 66549 30026<-01)
-4.36808 34364 41685(-0l)
-8.57092 52955 24884(-01)
-3.92465 30179 95107(-0l)
8.60212 23900 26319(-01)
1.86895 58009 44782(-04)
-9.11142 60410 63340(-03)
1.06097 02731 82322(-01)
-4.88842 63971 64796(-01)
1.00000 00000 OOOOO(-OO)
-7.08369 80999 49995(-0l)
-4.70608 86461 34285(-01)
9.44390 21369 08030(-01)
-3.73891 52081 99850(-01)

ORDER OF MATRIX * 9 (CONT.)
EIGENVALUES
3.49967 04029 ll493<-12>
EIGLNVECTORS
1.36620 49070 13275(-05)
•9.47535 57566 95441(-04)
1.61058 61751 07205C-02)
-1.15383 19398 86663("01)
4.24473 99502 14224(-01)
-8.68875 53580 11983(-01)
1.00000 00000 OOOOO(-OO)
-6.05131 38110 13442C-01)
1.49754 18355 81289(-01)
ORDER OF MATRIX
10
EIGENVALUES
1.75191 96702 b5178(-00>
3.57418 16271 b3924<-02>
E]
1.00000
6.08991
4.53138
3.65286
3.07753
2.66725
2.35801
2.11563
1.92005
1.75860
-5.06044
1.00000
6.31415
2.40699
-4,58618
-2.45040
-3.82178
-4.76401
-5.4083b
-5.84363
[GENVEC
00000
91436
29895
01340
04744
18429
3079b
96395
12818
03439
64866
00000
38 757
24192
75111
59873
19752
86973
48066
76347
;tors
OOOOO(-OO)
96503(-01)
94215(-01)
21510(-01)
55016(-01)
30508(-01)
24843(-01)
1540K-01)
6119K-01)
31029(-01)
39978(-01)
OOOOO(-OO)
61648(-01)
57739(-01)
07758(-02)
37491(-01)
80346(-01)
73599(-01)
89185(-01)
9241K-01)
EIGENVALUES
3.42929 54848 35091(-01)
2.53089 07686 70038(-03)
EIGENVECTORS
1.00000
1.10465
3.66282
4.42425
60536
56341
43036
26171
08260
-3.90483
.83100
,00000
►50413
►15015
,09646
,39133
,36629
,55456
,10538
7.27979
00000
17177
37964
91767
94518
25730
22451
31611
80801
78675
74913
00000
96416
97273
75122
24904
40180
55324
49054
70808
OOOOO(-OO)
43785(-01)
87492(-01)
28277(-01)
17078(-01)
07357(-01)
70023(-01)
63669(-01)
73888(-01)
18817(-01)
05559(-01)
OOOOO(-OO)
78791(-01)
02427(-01)
58232(-01)
62540(-01)
83149(-02)
15149(-01)
81802(-01)
98943(-01)

ORDER OF MATRIX = 10 (CONT.)
tlotNVALULS
1.28749 01427 bi/71(-04)
1.22896 77J87 bll75(-07>
2.26674 67477 62926(-ll>
EIGENVECTORS
5.02691 93708 19130(-Q2)
5.5371b 19652 43531(-01)
1.00000 00000 OOOOO(-OO)
2.90771 16075 67017(-01)
3.93565 04576 91000(-01)
6.42818 76489 72120(-01)
5.30534 27508 17633(-01)
1.92318 51809 90450(-01)
2.64461 64931 15005(-01)
7.68640 20927 86739(-01)
2.20211 62655 20837(-03)
6.65308 34126 35937(-02)
4.50423 02451 95166(-01)
1.00000 00000 OOOOO(-OO)
3.97406 18263 63392(-01)
7.52617 34673 95069(-01)
5.19153 37204 50187(-02)
6.56558 24094 19764(-01)
5.76366 49/89 6l755(-01)
6.46988 57273 97804(-01)
3.73254 77290 76785(-05)
2.37327 10245 65077(-03)
3.64962 96430 35628(-02)
2.29661 97587 73360(-01)
6.96603 89892 21241(-01)
1.00000 00000 OOOOO(-OO)
3.68315 31552 82386(-01)
6.99899 14802 67557(-01)
8.48690 27725 93223(-01)
2.79403 11288 36095(-01)
EIGENVALUES
4.72968 92931 82348(-06)
2.14743 88173 50479(-09)
1.09315 38193 79666(-13)
EIGENVECTORS
1.34197 07196 31349(-02)
-2.56349 00053 68558(-01)
1.00000 00000 OOOOO(-OO)
-8.26254 49309 85651(-01)
•7.30398 87720 041901-01)
9.79799 70771 83957(-02)
6.90035 19449 49648(-01)
7.09559 50102 98084(-01)
1.52328 09663 07994(-01)
-8.64618 11259 32675(-01)
3.29136 10483 095l0(-04)
-1.47645 74515 12742(-02)
1.55750 54872 45585(-01)
-6.25397 96321 29557(-01)
1.00000 00000 OOOOO(-OO)
-2.34914 01869 53356(-01)
-7.83112 53616 02477(-01)
1.07843 33964 72780(-01)
8.66944 29913 73357(-01)
-4.72962 81021 99039(-01)
2.71471 31336 04098(-06)
-2.36061 26295 90383(-04)
5.05289 73867 16890(-03)
-4.61160 40049 98925(-02)
2.20661 51772 89104(-0D
-6.08176 78395 43368(-01)
1.00000 00000 OOOOOi-00)
-9.68158 87951 22191(-01)
5*09073 58516 7l383(-01)
-1.12104 94021 47474C-01)

74 Matrices for Testing Computational Algorithms
Example 4.14
n n-1 n-2
n-1 n-1 n-2
n-2 n-2 n-2
2
1
2
1
2 1
2 1
2 1
Eigenvalues:
\
-*G
1 - COS
(2i-l)
2n+l
*r.
i - 1, 2, • • •, n.
Characteristic poljmomial for n = 12:
P(A) = A12- 78AU+ X001A10- 50ff5A9+ 12870A8- 1S448A7* 18564A6- 11628A?
+ 4845A4 - ISSOA3* 231A2- 23A + 1
Inverse: Example 3.12
Reference: [24].
Example 4.15
A =
n
Eigenvalues:
n-1
1 2
A- = A0 = ... = A 0 = 1
1 Z n- L
n-1
1
2
n-1
n
\ - and A are the roots of A - (n+l)A + defc(A ) - 0 where
n-1 n N n'
det(A. ) . , n(n+l)(2n-5)
n o
Inverse: Example 3.16
Reference: [l].

Eigensystems—Real Symmetric Matrices 75
Example 4,16
5
•4
1
Eigenvalues:
-4
6
-4
1
-4
6
1
-4
■4
1
6
-4
1
-A
6
■4
(2(11+1);'
Reference: [18, p. 20], [58]
n x n.
7^ = 16 sin [ 0 /^A1 ^ J , k = 1, 2 n.
Example 4.17
A -
where
(B+I ) 161
n n
161
n
B =
B
161
161
n
B 161 -I
•I 161
n n
B
161
n x n
161
n
(B+I )
59
16
-1
16
-60
16
-1
16
-60
-1
16
-1
-1
16
-1
-60
16
16
-59
n x n.

76 Matrices for Testing Computational Algorithms
Eigenvalues of A:
A = t. + t - 60, i,j = 1, 2, ..., n, where
Reference: [36, pp. 22-24],
= 66 - (8 + 2 cos ^)2 ,
Example 4.18
where
A =
X
Y
Y
X
X =
■20
4
Y
Y
4
■20
X
Y
4
4
Y
X
2 2
n x n
-20 4
4 -20
n x
and
Y =
4
1
1
4
1
1
4
1
1
4
n x II.
Eigenvalues of A:
A = ('20 - 8 cos k9 - 8 cos j9 + 4 cos k9 cos j6); 9
k,j
Reference: [36, pp. 22, 24].

Eigensystems—Real Symmetric Matrices 77
Example 4.19
A =
-41
r
X
X
-41
n
X
-41 X
n
X -41
n
2 2
where
X =
0
1
1
0
0
1
1
0
, n x n.
Eigenvalues of A:
A,. = -4(l+cos k© cos j9); 9 ■-t-t , and k,j = 1, 2, .. ., n.
Reference: [36, pp. 22, 24].
Example 4.20
-1 2a
2a 0
1 2a
1
2a
0
1
2a
2a
1
0
2a
2a
-1
n x n.
Eigenvalues
^k
A, = |a-2 cos —r\ - fa
"(•
n+1
;
(a +2), k = 1, 2, ..., n.
Reference: [18, p. 31].

78 Matrices for Testing Computational Algorithms
Example 4.21
Let p be a prime, p ^> 5, and let n = p - 1. Define A = [a. J to be the
n x n matrix such that
Eigenvalues:
0, if p |(i+j)
a..= <. 1, if i + j i$ congruent to a square mod p
•1, otherwise
At = \fp9 i = 3, 4, ...,-+ 1
\ ~ "VF i = "o + 2> • • • > n*
Inverse; Example 3.14.
Reference: [43].

Eigensystems—Real Symmetric Matrices 79
Example 4.22
Let A = [a..] be the n x n matrix defined by
*ij " I" JI
A has a dominant positive eigenvalue and n-1 real negative eigenvalues.
If n = 2 (mod 4), then X = -1 is an eigenvalue with corresponding
eigenvector
1
-1
x = | -1
1
where the four components shown are repeated periodically.
Inverse: Example 3-23
Reference: [77, pp. 32-33]

80 Matrices for Testing Computational Algorithms
Example 4-23 (see Chapter 2, Section 4)
Let J be the n x n matrix all of whose elements are 1. Let
nn
f denote the column vector all of whose components are 1. Thus
nn
111
111
111
111
and f =
n
X = n is a simple eigenvalue and f is its corresponding eigenvector.
n
n
X- = X. = ... = X - = 0 and every vector orthogonal to f is an eigen-
1 Z n-1 n
vector corresponding to X = 0. In fact, the n-1 dimensional subspace of
eigenvectors corresponding to X = 0 is spanned by
g. = f - ne ,
&i n n
1=1, 2, ..., n-1
where e is the column vector whose components are 8.., j = 1, 2, . . *, n.
Reference: [7], [77]

CHAPTER V
TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF REAL NONSYMMETRIC MATRICES
Example 5.1
33
24
-8
16
-10
-4
72
-57
-17
Eigenvalues:
Right Eigenvectors:
A1=l
A^ - 3
-15
12
4
9
X2 =
-16
13
4
■4
3
1
Left Eigenvectors:
yx = [l, o, 4]
y2 = [0, 1, -3]
y3 - [4, 4, 3]
Inverse: Example 3.1
Reference: [29, pp. 65-69]
81

82 Matrices for Testing Computational Algorithms
Example 5.2
4 1
2 4
0 1
1
1
4
Eigenvalues:
Right Eigenvectors:
*1 = 3
A2 = 3
A3 = 6
x, =
0
1
-1
x„ =
3
4
2
Left Eigenvectors:
yx = [2, -1, -1]
y3 - [i, i, l]
Note: Corresponding to the multiple eigenvalue A- = A~, we have only
dimensional subspaces of right and left eigenvectors since the matrix
defective, x- and y- are vectors from these subspaces.
Reference: [54],

Eigensystems—Real Nonsymmetric Matrices 83
Example 5.3
1 0 0.01
0.1 1 0
0 11
Eigenvalues:
\ = 1 + O.lo) where a) is a cube root of unity, i.e.,
0) €
Hfr**)}
Right Eigenvectors:
x =
0)
0)
10o)
Left Eigenvectors:
y^ = ,[1, a), O.la) ]
Reference: [17]

84 Matrices for Testing Computational Algorithms
Example 5.4.
8
-4
18
-1
4
-5
-5
-2
-7
Tridiagonal Eorm from Lanczos* Method;.
a.
^
291
43
162
43
-86
23120
1849
Eigenvalues:
AL = 2 + 4i
A2 = 2
A3 = l
4i
Right Eigenvectors:
x. =
l
1-i
2
-2i
x„ =
1+i
2
2i
Left Eigenvectors:
yx = [10, -3-i, -4+2i]
y2 = [10, -3+i, -4-2i]
y3 = [2, -1, -1]
Reference: [22, pp. 256-257].

Eigensystems—Real Nonsymmetric Matrices 85
Example 5.5
■2
■3
■2
•1
2
3
0
0
2
2
4
0
2
2
2
5
Eigenvalues:
Right Eigenvectors:
V1
A2 = 2
A3 = 3
A4 = 4
14 1 13 1 2
3 3 2
X1 — I I » X9 — I I ' X^ I
2 2 2
111 111 I 1
Left Eigenvectors:
yi
Vo =
y,. =
[1, -1, 0, 0]
[-1, 2, -1, 0]
[0, -1, 2, -1]
[ 0, 0, -1, 2]
x, =
Reference: [15].

86 Matrices for Testing Computational Algorithms
Example 5.6
Eigenvalues:
6 -3
4 2
4 -2
4 2
X, =
X„ -
X„ =
X, =
3 +\/5
3 +V^5
3 - \fl
3 - \/5
4
4
3
3
1
0
1
1
Right Eigenvectors:
Xl =
3 + Vi
2
6
X3 =
-\/5
3 - \fl
2
6
Left Eigenvectors:
yt = [5 + \/5, -(5 +y/5 ), 3^/5 /2, 5/2]
y3 = [5 - \fl , -(5 - \fl ), --H/r/z, 5/2]
Note: Corresponding to the multiple eigenvalue X. = A_, we have only one-
dimensional subspaces of right and left eigenvectors since the matrix is
defective. x1 and y1 are vectors from these subspaces. Similarly, x. and y,
are vectors from the one-dimensional subspaces of eigenvectors corresponding
to the multiple eigenvalue A„ = A,.
Reference: [17].

Eigensystems—Real Nonsymmetric Matrices 87
Example 5.7
Eigenvalues:
~ 0
1.31
1.06
-2.64
h
\
0.07
-0.36
2.86
-1.84
= 0.03
= 3.03
= -1.97
= -1.97
+
-
0.27
1.21
1.49
-0.24
i
i
-0.33
0.41
-1.34
-2.01
References: [21],
Example 5.8
Eigenvalues:
4
0
5
3
h'
X2 =
X3 =
X, =
-5
4
-3
0
12
1 + 5i
1 - 5i
2
0
-3
4
5
3
■5
0
4
Right Eigenvectors:
111 1
_ -1 _ -i
Xl " . ' X2 ~ .
I 1 I -i
1 -1
x„ -
1
i
i
•1
x, =

88 Matrices for Testing Computational Algorithms
Left Eigenvectors:
y]_ = [1, -1, 1, 1]
y2 = Ci, i. i, -i]
y3 = [1, -i, -i, -1]
y4 = [1, 1, -1, 1]
Reference: [49, pp. 57-58], [60, p. 147],
Example 5.9
Eigenvalues:
122
40
27
| 32
r
41
170
26
22
28
40
25
172
9
-2
26
14
7
106
-1
25
24
3
6
165
A, = 242.97727 3320
A„ = 167.48487 8917
K = 134.68646 3320
A, = 112.15419 3247
Ar = 77.69719 11963
Reference: [38].

Eigensystems—Real Nonsymmetric Matrices 89
Example 5.10
0.4163
0.0001
0.6321
0.5157
0.5563
0.3176
0.4132
0.3157
0.8321
0.4431
0
0.8175
0.4823
0.5642
0.2567
0
0
0.6614
0.6541
0.8325
0
0
0
0.4321
0.8475
Tridiagonal Eorm from the Elimination Method:
0.4163 0.3176
0.0001 5167.8307
0 -3265 9398.0809
0 0
0 0
0 0 0
0.8175 0 0
■5166.3956 7804 0.6614 0
-0.0001 1909 0.5615 0.4321
0 0.5463 0.4006
Eigenvalues:
^ = 1.8390
A2 = 0.2363
A3 ^ 0.8045
A4 =-0.0332 + 0.4374i
A5 =-0.0332 - 0.4374i
Reference: [69].

90 Matrices for Testing Computational Algorithms
Example 5.11
15
1
7
7
17
11
3
6
7
12
6
9
6
5
5
-9
-3
-3
-3
-10
-15
-8
-11
-11
-16
Characteristic Polynomial:
P(A) = A5 - 5A4 + 33A3 - 51A2 + 135* + 225
Eigenvalues:
A, =
K -
A. =
A, =
A. =
1.5 +\/l2.75 i
1.5 + /12.75 i
1.5 - \/l2.75 i
1.5 - 1/12.75 i
-1
Right Eigenvectors:
xl=
184
507
295.5
411
213.5
- 230 /12.75 i
- 52 \/i2.75 i
- 163 >/l2.75 i
- 166 \/l2.75 i |
- 223 \/l2.75 i
• X3 =
184 + 230/12.75 i
507 + 52^12.75 i
295.5 + 163^12.75 i
411 + 166^12.75 i
213.5 + 223/12.75 i
» Xc
Left Eigenvectors:
yx = [l50 - 342/12.75 i, 184 - 230/12.75 i, -1630 - 74/12.75 i,
589.5 + 409/12.75 i, 555 + 156/12.75 il

No
Eigensystems—Real Nonsymmetric Matrices 91
y3 = fl50 + 342/12.75 i, 184 + 230^12.75 i, -1630 + 74^12.75 i,
589.5 - 409\^12.75 i, 555 - 156^12.75 i]
y5 = [-25, -3, 16, -7, 20]
te; Corresponding to the multiple eigenvalue A- = A9, we have only one-
dimensional subspaces of right and left eigenvectors since the matrix is
defective, x- and y~ are vectors from these subspaces. Similarly, x~ and y-
are vectors from the one-dimensional subspaces of eigenvectors corresponding
to the multiple eigenvalue A and A..
Reference: [15], [51], [60, pp. 143-144].
Example 5.12
10
9
8
6
4
2
-19
-18
-16
-12
-8
-4
17
17
15
12
8
4
-12
-12
-11
-10
-6
-3
4
4
4
4
1
1
1
1
1
1
2
0
Eigenvalues:
v-
*6"
-1
•1
■1
1
Reference: [58].

92 Matrices for Testing Computational Algorithms
Example 5.13
where
B =
with € = 10
Eigenvalues of A:
0
0
0
0
e
A =
1
0
0
0
0
[~B 2B~|
4B 3Bj
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
A = 0.5 exp (2k7Ti/5), k =
A =-0.1 exp (2k7Ti/5), k =
Reference: [45], [61],
Example 5.14
Eigenvalues:
12
11
10
2
[_1
11
11
10
2
1
10
10
2
1
9
2
1
2
1
= 32.22889 15015 72160 750
= 20.19898 86458 77079 428
= 12.31107 74088 68526 120
= 6.96153 30855 67122 113
= 3.51185 59485 80757 194
= 1,55398 87091 32107 90

Eigensystems—Real Nonsymmetric Matrices 93
A? = 0.64350 53190 04855 5
Ag A 0,28474 97205 58478
A9 = 0.14364 65197 69220
A1Q = 0.08122 76592 40405
Au = 0.04950 74291 85278
A12 = 0.03102 80606 44010
We give the right and left eigenvectors corresponding to Ain, A.. , A19,
the three most sensitive eigenvalues. See Varah [79, pp. 107-111] for additional
vectors.
Right Eigenvectors:
X10
-0.67714
0.73369
-0.05625
-0.00084
0.00060
-0.00006
0.00000
0.00000
-0.00000
0.00000
0.00000
0.00000
Left Eigenvectors:
yio
0.00000
0.00000
0.00000
-0.00000
0.00001
-0.00002
-0.00068
0.00946
-0.06504
0.26984
-0.64994
0.70740
11
91
42
53
06
06
09
07
01
00
00
00
00
00
00
10
10
37
17
06
47
68
86
99
Xll
-0.67599
0.73440
-0,06061
0.00211
0.00011
-0.00002
0.00000
-0.00000
0.00000
0.00000
0.00000
0.00000
yu
0.00000
0.00000
0.00000
0.00000
-0.00001
0.00021
-0.00221
0.01596
-0.08251
0.29459
-0.65581
0.68997
46
62
69
69
26
68
29
02
00
00
00
00
00
00
00
03
41
95
72
10
35
26
13
00
X12
-0.67531 89
0.73480 66
-0.06315 65
0.00385 87
-0.00019 87
0.00000 91
-0.00000 04
0.00000 00
0.00000 00
0.00000 00
0.00000 00
0.00000 00
y12
0.00000 00
0.00000 00
-0.00000 03
0.00000 38
-0.00004 51
0.00043 20
-0.00331 87
0.02009 76
-0.09284 49
0.30854 36
-0,65861 25
0.67970 23

94 Matrices ioi Testing Computational Algorithms
Condition Numbers are |s.| , where
1
s = 0.30424 083
s2 = -0.20079 033
s, * 0.31822 599
s. = -0.58447 355
4
s = 0.14446 703
sr = -0.00462 656
S7
S8
S9
10
11
= 0.00006 913
= -0.00000 178
= 0.00000 01498
= -0.00000 00375
= 0.00000 00258
= -0.00000 00547
References: [17], [71], [72, pp. 151-153], [79, pp. 106-111]
Example 5.15
38747
-49239
-125005
-175215
-176459
-68786
-70392
-66818
-56793
-39309
-15085
-107826
-30063
-10446
-31693
-74761
-9747
-65257
-59161
-60822
162836
147007
.. 135534
127430
125487
129710
-12116
-3369
-1163
-3527
26624
523895
760217
707955
356979
13606
74578
36906
-7997
-47171
-76321
-54142
-15101
-5252
-15937
-51553
94918
-20436
78513
62926
110162
696799
690486
682860
675886
670239
-1960
-547
-190
-578
46397
874986
1889996
1585904
1421590
-1635
-20029
-35373
-190532
-336666
-459117
-47535
-13276
-4631
-14052
-49730
41968
-30683
149178
121436
101956
645055
1247127
1233574
1220769
1209950
-948
-266
-93
-284
21041
518535
987939
1835031
1490020
5801
32720
'66689
121047
-92801
-276532
-18910
-5304
-1871
-5675
-39456
-15257
-157418
188162
152216
76674
485251
938227
1542211
1526106
1512267
157
42
12
38
10983
207320
812227
1452154
2432634
1970
11970
28893
65580
165443
-166380
-4472
-1304
-501
-1523
-41365
-67307
-305659
-110555
225707
75926
480500
929431
1528044
2189518
2169459
1128
311
105
320

Eigensystems—Real Nonsymmetric Matrices 95
-34389
-105605
-390172
-389762
-285526
58807
372486
720265
1184459
1698300
2369651
1869
518
177
536
-16813
-182666
-363323
-400208
-358272
. 11793
23216
41433
70143
108178
153437
1536234
285553
-18787
-57006
-50316
-595834
-1185925
-1327062
-1202147
30018
49257
95189
178721
298506
448574
18473
108570
-58126
-176348
-40743
-527558
-1044921
-1183861
-1079850
19232
15034
38606
97822
194757
324414
-519690
-144914
-932
12799
-31954
-439963
-863211
-979778
-890416
11895
-6149
2197
41610
115175
217277
-438975
-122917
110583
317581
Eigenvalues:
\
h
h
\
\
\
h
\
\
\o
\l
X12
X13
A14
*13
•
5=
•
=
•
•
£
=
=
=
•
=
&
=
=
6294127.73
4830173.88
1593256,12
1296443.15
976578.%
517836.12
369921.48
308201.60
257611.17
173583.27
151487.87
73704.19
43990.81
3587.14
-605.47

96 Matrices for Testing Computational Algorithms
Right Eigenvectors:
*1 *2 *3 *4 *5
-25600791
-93333619
-281658289
-234339803
-180185122
41356495
255125017
466609433
693438409
876398341
1000000000
5717639
1491503
425148
1287435
8179865
281884032
618796420
892390433
1000000000
13057924
78558762
136928986
179243759
166202576
51289932
-19636264
-5022691
-1324500
-4008552
-10559890
-154642407
-135528630
172711194
567634488
1504059
-37368379
-47836788
-16468980
18990089
-111765187
1000000000
202341288
-23478411
-70672023
-6382980
-514788753
-635147361
-51309046
1000000000
-42915249
-228515241
-275319457
-90455002
200223576
39001492
-154945349
-28449817
10379135
31184833
71337697
230978290
463951989
-149941032
-79550827
-111068725
-582184747
-737157698
-393746553
310698214
1000000000
38153292
5665174
-6683085
-20016569
^6 ^7 ^8 ^9 ^10
35546791
18937820
-430319318
1000000000
-469843941
-81892147
-353035016
-211843343
425924519
-37494408
-30655114
-17087
8560
340161
1016943
-96033014
88496756
-381761741
622739632
65080697
195869218
662203426
-12939 707
-716871720
-632446333
1000000000
15840072
-2494823
15190410
44562429
-43838567
114088163
584957693
189896517
42051926
74731927
-221204146
17654019
163015274
-16984885
-48074780
532263383
-150431718
343042433
1000000000
-125099214
590391245
-214921796
85285698
-147661918
343989557
777971729
-957697355
-102833479
1000000000
-706623434
35799075
-16577034
17464795
50539011
-316813407
-374763409
302668034
-282248681
117044666
605390857
84552451
-568699623
1000000000
-688507740
260312429
-40233177
54361072
-9100840
-25757337
Xll *12 *13 *14 *15
-483941289
-54509219
-25563085
294522536
-179052534
1000000000
-718306625
560841987
-700735268
509633733
-167348227
-33933888
71277033
-2855789
-7992436
80100044
1000000000
-683069341
-560808914
319086961
329734642
-326232825
140258619
307477539
-361747416
109768320
-88089124
-367241747
-96270976
-251404803
158500974
1000000000
217439196
135628883
331963034
-211818796
-94574848
29608774
78272228
-140465927
-7629423
291730040
646206560
231004723
556214353
1000000000
457078895
509694713
387188842
208187123
187142783
10731580
-13901773
-29683858
-14317443
-38506599
403471483
576215354
435241146
536274143
926568323
941981992
817688917
609029854
382268933
-26357098
-11586613
-18513917
-32489449
-47897332
-62329715
686089278
947542071
1000000000
795951845
Reference: [73],

Eigensystems—Real Nonsymmetric Matrices 97
Example 5.16
B 2B
4B 3B
]
where
and
C =
B =
-2
-3
■2
-1
5C
5C
2 2
3 2
0 4
0 0
]
Eigenvalues of A:
XL - 15 + 5 i
X2 = 15 - 5 i
X = 30 + 10 i
X. = 30 - 10 i
4
X5 = 45 + 15 i
X, = 45 - 15 i
6
Xy = 60 + 20 i
Xg = 60 - 20 i
X10 =
11
12
13
14
\5 =
16
3 + i
3 - i
-6 + 2 i
-6 - 2 i
-9 + 3 i
-9 - 3 i
-12 + 4 i
-12 - 4 i
Reference: [15].

98 Matrices for Testing Computational Algorithms
Example 5.17
A =
where
B =
C =
D -
8B
-5B
rc
L4c
"4D
r2D
-1
-1
-1
-1
-1
4B~
-B_
2C~
3C _
3D~
-D -
1
0
0
0
0
1
!•
1
1-
1
1-
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
Eigenvalues of A:
8, 6, 4, 3, -40, -30, -20, -15, ±40j, ±30j, ±20j, ±15j,
±8j, ±.6j, ±4j, ±3j where j = \ fl ± & l) .
Reference: [15], [45], [61].
Example 5.18
where
■C
B
C =
B
4B
3C
5C
6D
8D
2D
5D
2B
3B_
3C_
C .
-D
0
0
-D
1
I'
D
D
D
-D
0
2D
2D
D

Eigensystems—Real Nonsymmetric Matrices
99
D =
■2
-3
-2
•1
2 2
3 2
0 4
0 0
2
2
2
5
Eigenvalues of A:
120j, 90j, 60j, 30j, -40j, -30j, -20j, -10J, -24j, -18j,
-12j, -6j, 8J,, 6j, 4j, 2j where j e {3 ± i, 1 + 2i}.
Reference: [45]
Example 5.19
Let A = [a..] be the 100 x 100 matrix defined by
/ 101 - i, if j = i
(_L)i+j+l 4o/(1:fj_2)f if j < i
a..
13
= < 40/102,
40,
if i = 1 and j = 2
If 1 ■ 1 and J ■ 100
otherwise.
V
Eigenvalues: erf At
1, 2, 3, ..., 99, 100.
Reference: [45].

100 Matrices for Testing Computational Algorithms
Example 5.20
Let A = [a..1 be the n x n matrix defined by
n ij
a =* 1, j =1, 2, . . ., n,
a±j = (i+j-1)"1, i = 2, 3, ..., n, j = 1, 2, .... n.
A =
n
1
2
1
3
I
3
1.
4
1
4
1
5
1
n+1
1
n+2
1
n+1
1
n+2
2n-l
Eigenvalues:
Let X__(n) and X (n) be the eigenvalues of A of largest and smallest
Mm n
magnitude respectively.
Vn>
-0.115 0693
n
2
3
4
5
6
7
8
9
10
Vn)
1.448 403
1.707 105
1.886 632
2.022 999
2.132 376
2.223 362
2.301 055
2.368 717
2.428 554
-0.481 5399 x 10
-2
-0.144 1324 x 10
■0.448 9833 x 10
-0.139 7499 x 10
■0.433 6577 x 10
-0.134 0623 x 10
-3
-0.412 9309 x 10
•11
-0.126 7649 x 10
-12

Eigensystems—Real Nonsymmetric Matrices 101
Right eigenvector corresponding to ^M(n);
n = 2
1.000000 1.
0.448403 0.
0.
7
1.000000
0.351545
0.251183
0.197135
0.162893
0.139091
0.121514
3
000000
416793
290313
1.
0.
0.
0.
0.
0.
0.
0.
1.
0.
0.
0.
8
000000
341809
244943
192700
159546
136461
119387
106209
4
000000 1.
394224 0.
277320 0.
215088 0.
0.
9
1.000000
0.333408
0.239479
0.188772
0.156554
0.134092
0.117457
0.104603
0.094352
Right eigenvector corresponding to X (n):
n = 2
-0.896805 0
1.000000 -1
1
7
0.008795
-0.176722
1.140790
-3.339147
4.912246
-3.545962
1.000000
3
.550163
.552812
.000000
-0
0
-0
2
-5
6
-4
1
-0
1
-2
1
8
.002721
.073168
.638751
.598579
.607667
.623060
.045669
.000000
4
.224188 0
.270271 -0
.046050 2
.000000 -2
1
9
0.000819
-0.028430
0.322286
-1.732159
5.099020
-8.699495
8.583374
-4.545416
1.000000
5 6
000000 1.000000
376917 0.363036
266965 0.258417
208099 0.202206
171019 0.166679
0.142038
10
1.000000
0.326051
0.234636
0.185257
0.153855
0.131940
0.115694
0.103129
0.Q93099
0.084893
5 6
.081470 -0.027443
.772313 0.392118
.237419 -1.768987
.546576 3.450596
.000000 -3.046283
1.000000
10
-0.000242
0.010516
-0.150100
1.027660
-3.934520
9.038123
-12.739602
10.793366
-5.045200
1.000000

102 Matrices for Testing Computational Algorithms
Eigenvalues of A.
V*
*6*
2.1323 763
-0.2214 0681
-0.3184 3305 x 10"
-0.8983 2330 x 10
-0.1706 2788 x 10
-0.1397 4990 x 10*
-3
-4
Inverse: Example 3.9
Reference: [32], [33],
Example 5.21
A -
■1
1
0
-1
0
1
-1
0
0
■1
0
0
-1
0
0
n x n.
Eigenvalues-:
> Stent
k = e35P :n+I~ » = ' ' " *"' ■n"
Right Eigenvectors;
(k)
Let x he the right eigenvector of A corresponding to the eigenvalue
A,, k = 1, 2a ..., n. Then

Eigensystems—Real Nonsymmetric Matrices 103
n-l
k
n-2
,(k>
Left Eigenvectors:
\y\ » y\ » ■••» y I be the left eigenvector of A
Let y'
corresponding to the eigenvalue 7v, k-* 1, 2, . .♦, n. Then, for each k,
?
vm
(k) _ m~0
Reference: [17].
cj
j = 1, 2, .. ., n.
Example ~5.22 (Forsythe)
0 1
0
0 .1
1
0
Characteristic Equation:
,n
A" - e = 0

104 Matrices for Testing Computational Algorithms
Eigenvalues:
nr—y 2j.
\ " V | e| exp -^— , k - 1, 2, ..,, n.
Reference: [61], [76], [62, p. 64].
Example 5.23
20 20
19 20
18 20
2 20
Characteristic Equation:
(20-A)(19-A) ... (1-A) - 20 e = 0
Eigenvalues:
If e = 10 , the eigenvalues are
0.99575439 3.96533070 ± 1.08773570i
20.00424561 17.03466930 ± 1.087735701
2.10924184 5.89397755 ± 1.948529271
18.89075816 15.10602245 ± 1.948529271
2.57488140 8.11807338 ± 2.529181731
18.42511860 12.88192662 ± 2.52918173i
10.50000000 ± 2.73339736i
Condition Numbers:
If e = 0, |s, | = (2°-k>,-(k-l>' , k = 1, 2, . . ., 20, and |s. f1 is the condition
k 20iy k
number corresponding to the eigenvalue A = k.
Reference: [62, pp. 90-91].

Eigensystems—Real Nonsymmetric Matrices 105
Example 5.24
For arbitrary constants a-, a0, ..., a - , let
1 L n-1
An " An(al,a2'*,''an-1)
" [aij]
denote the n x n matrix defined by
lj
J < i.
For example,
\ =
1
1
If we define
P (X) = (X-l+a^ (X-l+ap ... (X-l+a ), j = 1, 2, ..., n-1,
we can write the characteristic polynomial of A as
xn - xn_1 - \n-\(x) - xn_3P2(x)
Pn-l<X>
If X ^ 0 is an eigenvalue of A and if x is a right eigenvector
corresponding to X, it can be shown that

106 Matrices for Testing Computational Algorithms
X =
n
where
XJ = X""(J"1)pj-i<A>» j = 2, 3, ...„n.
If A # 0 isi a multiple eigenvalue of A^, there is only a one-dimensional
.subs^aee of eigenvectors associated with Ay, i.e., A is defective.
If A = 0 is am eigenvalue of A , the expression for det(A ) shows
n
n
that a. = 1 for certain values of i. Suppose that A = 0 is an eigenvalue of
multiplicity k and that a. = ... = a. = 1. Then there is a k-dimensional
h \
subspace of eigenvectors associated with A = 0, and the vectors
e - e. , j' - 1, 2, . . . , k,
forma basis for the subspace.
Inverse: Example 3.24.
Reference: [75]

Eigensystems—Real Nonsymmetric Matrices 107
Example 5.25
A =
Eigenvalues:
1
' 3
4
3
4
" 3
4
" 3
4
" 3
4
" 3
1
6
2
3
5
6
5
6
5
6
5
6
0
0
1
-1
-1
-1
9
2
9
27
2
39
2
43
2
43
2
- 3
- 6
■^ - 9
^ -12
=£ "13
^ -10
-4
-5
Xl =
X2 =
•3
-2
X3"
X4"
3_
2
X5 =
X6 =
I
3
Right Eigenvectors:
x, = —
111 111 II
I 2 I 2 I 2
13 -13 -I3
6 4' X2"54' X3 44
I 5 I I 5 I I 4
6 I 5 I I 4

108 Matrices foi Testing Computational Algorithms
1
X4=3
1
2
3
3
3
3
x_ = -r
1
2
2
2
2
2
x, =
1
1
1
1
1
1
Reference: [79, p. 100]
Example 5.26
A =
9
10
8
6
4
2
21
21
16
12
8
4
-15
-14
-11
- 9
- 6
- 3
4
4
4
3
0
0
2
2
2
3
5
1
0
0
0
0
0
3
This matrix is defective and also derogatory [80]. It has an eigenvalue
of multiplicity 2 corresponding to a quadratic elementary divisor and an
eigenvalue of multiplicity 2 corresponding to two (equal) linear elementary
divisors. The other two eigenvalues are complex conjugates.
Eigenvalues:
X, = X„ =
x« =
X, =
X„ = X, =
3
2 + i
2 - i
1
(linear elementary divisors)
(a quadratic elementary divisor)

Eigensystems—Real Nonsymmetric Matrices 109
Eigenvectors:
Corresponding to X = X we have the two-dimensional subspace of
eigenvectors spanned by
xi=
i
i
i
i
i
0
X2 =
0
0
0
0
0
1
Corresponding to X, and X, we have x, and x, given by
-1:
61
we hi
61
5(ll±i)
4(ll±i)
3(ll±i)
2(ll±i)
(ll±i)_
ive the eij
genvector
Corresponding to X = X we have the eigenvector X- and the principal vector
of degree two, x,, given by
o
i
X5 "4
4
4
4
3
2
1
1
X6 = 3
3
3
3
3
2
1
These six normalized vectors (five eigenvectors and a principal vector) make
up the transformation to Jordan canonical form.
Reference: [79, p. 103 and p. 207].

110 Matrices for Testing Computational Algorithms
Example 5.27
2
4
6
8
10
12
15
12
12
12
-2
-4
-6
-8
-10
-12
-13
-11
-14
-14
4
8
12
16
20
24
28
32
37
36
-3
-6
-9
-12
-15
-18
-21
-24
-26
-25
This matrix is both defective and derogatory [80]. As a matter of fact, a
multiple eigenvalue is associated with more than one nonlinear elementary
divisor in two instances. To be specific, X = 2 is an eigenvalue of
multiplicity 5 but it is associated with two nonlinear elementary divisors, one
of degree 3 and one of degree 2. Likewise, X = 3 is an eigenvalue of
multiplicity 4 but it is associated with two quadratic elementary divisors.
Consequently, the results are grouped according to the invariant subspaces
spanned by the eigenvectors and principial vectors shown.
Eigenvalues, Eigenvectors,, and Principal Vectors:
X- = X = X = 2 is associated with the eigenvector X- and the principal
vectors x and x of degrees 2 and 3, respectively, given by
A =
1-1
1-1
1 2
1 0 •
1 0
1 0
1 0
1 0
1 0
•I 0
1 0
1
3
5
3
3
3
3
3
3
3
-2
-4
-5
-4
-6
-6
-6
-6
-6
-6
1
2
3
4
5
2
2
2
2
2
-1
-2
-3
-4
-4
-2
-5
-5
-5
-5

Eigensystems—Real Nonsymmetric Matrices 111
xi =
x„ = —
1
2
2
2
2
2
2
2
2
2
x„ = —
1
2
3
3
3
3
3
3
3
3
X = X, = 2 is associated with the eigenvector x, and the principal vector
Xp of degree 2 given by
1
X4=4
1
2
3
4
4
4
4
4
4
4
1
X5 "5
1
2
3
4
5
5
5
5
5
5
= X = 3 is associated with the eigenvector xfi and the principal vector
X, =
x_ of degree 2 given by

112 Matrices for Testing Computational Algorithms
X, = -7
1
2
3
4
5
6
6
6
6
6
x, = —
1
2
3
4
5
6
7
7
7
7
X. = X = 3 is associated with the eigenvector x„ and the principal vector
x of degree 2 given by
1
X8=8
1
2
3
4
5
6
7
8
8
8
x^ = —
1
2
3
4
5
6
7
8
9
9

Eigensystems—Real Nonsymmetric Matrices 113
Finally, X- = 1 is associated with the eigenvector
T.0
1
10
l~l
2
3
4
5
6
7
8
9
10
These ten normalized vectors (five eigenvectors and five principal
vectors) make up the transformation to Jordan canonical form.
Reference: [79, pp. 211-212].
\

CHAPTER VI
TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF COMPLEX MATRICES
Example 6.1
Li 1J
Eigenvalues:
Eigenvectors:
*!-[;]. ■*-[■;]■
Reference: [40, p. 142].
Example 6.2
& ?]
Eigenvalues:
Ax = 1 + \ft
A2 = 1 - \fl
Eigenvectors:
Reference: [40, p. 142].
114

Example 6.3
Eigensystems—Complex Matrices
115
Eigenvalues:
2 -i 0
i 2 0
0 0 3
\~i
X2 = 3
Eigenvectors:
x, =
-1
i
0
x„ =
1
i
0
x„ =
Note: Corresponding to the multiple eigenvalue A- = A~, we have a two-
dimensional subspace of eigenvectors. x~ and x« are two orthogonal vectors
from this subspace.
Reference: [40, p. 99].
Example 6.4
Eigenvalues:
1+2 i 3+41
43+44i 13+141
5+6i 7+8i
Ax = 6.70088 -
21+221
15+161
25+26i
7.87599 i
A2 = 39.7767 + 42.99567 i
A3 = -7.47753 + 6.88032 i
Reference: [15].

116 Matrices for Testing Computational Algorithms
Example 6.5
5 + 91
3 + 3i
2 + 21
1 + 1
5+51
6 + 101
3 + 3i
2 + 2i
-6 - 61
■5 - 5i
■1 + 3i
■3 - 3i
-7 - 7i
-6 - 6i
-5 - 51
41
Eigenvalues:
A, -
A„ =
A, =
1 + 51
2 + 6i
3 + 71
4+81
x_ =
Right Eigenvectors:
2
1
1
1
Reference: [60, p. 153].
|11 [~-l
2 x = "l
I ' 3
1 J 0
1 -1
x, =
-1
-1
■1
0
Example 6.6
Eigenvalues:
3
1
0
-2i
1
3
2i
0
0
-21
1
1
2i
0
1
1
A, =
A, =
2 + 2 \fl
2 - 2 \fl
4
0

Eigenvectors:
Eigensystems—Complex Matrices 117
xi =
1 +\/2
1 + yfl
i
■i
x„ =
1
1
(1+^2 )i
(1+V^ )i
x„ =
f-i~i r 11
i -i
1 I ij
Reference; Constructed from Examples 6.1 and 6.2 using similarity
transformations. See Ortega [44],
Example 6. 7
Eigenvalues
r 7
3
1 -
__-l -
.
2i
2i
1
-1
3
7
+ 2i
+ 2i
1 + 2i
1 - 2t
7
-3
-1 + 2i
-1 - 2i
-3
7
~h± - 0
\ = 8
A3 = 8
\=12
Eigenvectors:
Xl =
|-1 1 [~-l + i~| i 1
1 .0 _1 , 1
-i Zl ° *
I -i I l \} + *] L1
Note: Corresponding to the multiple eigenvalue A2 = A~, we have a two-
dimensional subspace of eigenvectors, x and x_ are two linearly independent
vectors from this subspace.
Reference: Constructed from Example 6.1 using similarity transformations.
See Ortega [44].

118 Matrices for Testing Computational Algorithms
Example 6.8
Let A = [a..] be the 5x5 Hermitian matrix with the following
elements:
i
1
*■
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
Eigenvalues:
J
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
a. .
Real Part
-8.45000 00000(-l)
5.20000 00000( 0)
3.01000 00000(-1)
-9.60000 00001( 0)
7.33999 99999(-2)
5.20000 00000( 0)
-6.20000 00000( 0)
-3.39000 00000( 0)
1.22000 00000(-1)
4.18999 99999( 0)
3.01000 00000(-1)
-3.39000 00000( 0)
1.90000 00000(-2)
9.35000 00000(-1)
-5.72000 00000(-2)
-9.60000 00001( 0)
1.22000 00000(-l)
9.35000 00000(-l)
7.21000 00000( 0)
3.37000 00000(-1)
7.33999 99999(-2)
4.18999 99999( 0)
-5.72000 00000(-2)
3.37000 00000(-1)
-1.23000 00000( 0)
Imaginary Part
0.00000 00000( 0)
1.03000 00000(-1)
-4.54000 00000(-2)
9.36000 00000(-l)
7.26000 00000( 0)
-1.03000 00000(-l)
0.00000 00000( 0)
-4.07000 00000(-1)
9.10000 00000(-l)
-3.66000 00000( 0)
4.54000 00000(-2)
4.07000 00000(-1)
0.00000 00000( 0)
-2.71000 00000(-1)
2.82000 00000( 0)
-9.36000 00000(-l)
-9.10000 00000(-1)
2.71000 OOOOO(-l)
0.00000 00000( 0)
6.03000 00000(-2)
-7.26000 00000( 0)
3.66000 00000( 0)
-2.82000 00000( 0)
-6.03000 00000(-2)
0.00000 00000( 0)
\ = 15.18016 5225
A2 i 5.67872 93543
*3 = -0.83398 68001 9
*4 = -5.14984 56282
A5 = -15.92106 2150

Eigensystems—Complex Matrices 119
Eigenvectors;
(k)
Let x. , j = 1, 2, 3, 4, 5, denote the components of the eigenvector
of A corresponding to the eigenvalue A,.
Real Part
x
(k)
Imaginary Part
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
6.00205
1.11536
8.01643
7.31287
4.60335
7.60360
3.20401
5.70992
2.88357
3.00383
3.64692
3.67168
3.10174
4.33126
1.45002
4.01976
3.75061
2.81651
2.75842
3.54373
5.82568
5.61740
6.64363
2.54317
1.10244
95562(
67096(
84119(
35545(
97946(
24995(
41313(
41394(
72893(
89915(
00756(
77161(
23865(
31670(
84663(
56816(
35332(
56461(
67208(
44505(
3754K
61319(
95437(
70782(
71280(
-1)
-1)
-3)
-1)
-2)
-2)
-1)
-1)
-1)
-2)
-1)
-1)
-1)
-1)
-1)
-1)
-1)
-1)
-1)
-1)
-1)
-1)
-2)
-1)
-1)
0.00000 00000( 0)
■1.01132 04497(-l)
4.44447 02492(-2)
-9.06611 51813(-2)
•2.64434 04390(-l)
0.00000 00000( 0)
2.35524 37576(-l)
-7.46531 68625(-2)
1.30852 91636(-1)
6.35064 97290(-l)
0.00000 00000( 0)
-5.51361 26644(-2)
5.24760 01509(-1)
-1.68050 17325(-1)
3.47419 94196(-1)
0.00000 00000( 0)
4.31499 42367(-l)
4.57399 51652(-1)
1.07759 84596(-2)
1.45685 63957(-l)
0.00000 00000( 0)
1.97319 27448(-l)
4.26394 75314(-2)
2.15142 03800(-3)
-4.72290 94206(-l)
Reference: [39].

120 Matrices for Testing Computational Algorithms
Example 6.9
1 + 2i
43 + 44i
5 + 6i
47 + 48i
9 + 101
Eigenvalues:
3 + 4i
13 + 14i
7 + 8i
17 + 18i
11 + 12i
21 + 22i
15 + 16i
25 + 26i
19 + 20i
29 + 30i
23 + 24i
33 + 34i
27 + 28i
37 + 38i
31 + 32i
41 + 42i
35 + 36i
45 + 46i
39 + 40i
49 + 50i
Ax = 127.38667 077303 + 132.27820 320006 i
X2 = 7.07331 324882 - 9.55838 903704 i
7\3 = -9.45998 402189 + 7.28018 583692 i
A. = 0.00000 000000 + 0.00000 000000 i
4
A„ = 0.00000 000000 + 0.00000 000000 i
Reference: [15], [26], [48].

Eigensystems—Complex Matrices 121
Example
2 +
3 +
5 -
2 +
1 +
5 -
5 +
-4 -
5
5 +
El
genva!
6.10
31
21
31
61
41
i
21
31
21
...
.ues:
3 +
-2 -
1 +
-2 +
2 +
1 +
7 +
2 +
2 +
0
0
0
0
1 +
7 +
-1 +
1 +
1 +
-7
1
i
21
31
21
41
41
31
21
61
6i
1
51
21
61
1
2
3
-3
1
6
1
1
1
4
3
1
1
3
0
+
+
-
+
+
-
+
+
-
0
0
0
0
0
-
+
+
—
21
i
i
71
51
51
61
31
31
21
41
21
31
-1
-4
1
-8
8
2
1
7
-4
6
2
5
0
0
+
+
+
-
+
-
+
+
0
0
0
0
0
0
+
+
+
■"
41
2i
5i
1
41
41
i
41
61
31
5i
41
5
2
4
4
3
-4
4
7
6
0
0
0
+
-
+
-
+
-
+
0
0
0
0
0
0
0
-
+
51
31
71
41
i
21
i
i
i
31
■ • *
0
0
0
0
0
0
0
0
3 + 2i
2 + 51
A = 4.16174868 + 3.137513561
A2 = 5.43644837 - 3.971425821
A3 = 2.38988759 + 7.268070711
A. i -1.93520144 - 3.975093821
4
A = -2.44755082 + 0.4371261751
A, = -5.27950616 - 2.275963031
o
A? = 1.03205812 + 9.294132781
AQ = -4.96687009 - 8.087124751
O
A = 8.81130928 + 1.54938266i
\1Q = 10.7976764 + 8.623381511
Reference: [65].

122 Matrices for Testing Computational Algorithms
Example 6.11
The matrix displayed on the following pages is a particular case
of a class of matrices sometimes referred to as the Dolph-Lewis matrices.
These matrices arise in problems in hydrodynamics and are described in [141.
The example included here is of order 20 and corresponds to a * 0,9• The
eigenvalues, eigenvectors, and condition numbers computed by Wilkinson [70]
are also given.
Notice that the diagonal elements of the matrix are complex numbers
but the off-diagonal elements are real. Wilkinson computed the eigenvalues
extremely accurately, and the complex eigenvalues are presented here to 31
significant digits.
Wilkinson also computed both the left eigenvectors and the right
eigenvectors along with the corresponding scalar products (condition
numbers) . These are included for completeness.

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A(I,J)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
-30816
11878
-12003
27916
59755
12347
-21178
63003
-22106
74032
-13349
-13404
25264
-30107
31502
-31175
30004
-28448
26751
-25043
REAL HART
64368
89436
89862
38560
48774
24996
48675
83422
84964
85781
08629
16201
82021
77436
19163
73942
92943
71851
57957
87266
51024
25986
06054
5*312
00398
61147
69804
05807
60614
03799
36014
85598
4*430
4*963
21711
29061
10156
26636
12824
52646
6277 (
5761 (
6875 (
8204 (
2544 (
5945 (
1916(
5666 (
3060 (
3729(
3095 (
7310(
2899 (
4690 (
6147(
9016<
2560 (
1773(
9ia9(
4373 (
0)
0)
0)
0)
0)
0)
-1)
-2)
-2)
-3)
-3)
-3)
-3)
-3)
-3)
-3)
-3)
-3)
-3)
-3)
imaginary part
-24625 98029 52408 7906<-]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20 1
r -28357
r 85823
r -57750
r 58531
f -86489
r 24549
r 59878
f 13785
r -26342
r 89195
r -37635
r 17503
r -83262
r 37223
r -12680
r -88497
r 84943
r -12725
r 14979
r -16047
98151
15314
03787
78212
89629
26613
40294
44121
18125
28336
90473
45996
01568
53831
17440
37014
42273
95400
15945
73697
79109
56112
13011
41915
00042
71827
83795
98066
60409
20419
68168
69743
70689
10819
45224
31590
4 7956
22350
94717
18137
5734( 0)
8616(-1)
7416(-1)
2260(-l)
5339(-l)
1255( 0)
1660< 0)
7U4( 0)
3075(-l)
3058(-2)
8309(-2)
3293(-2)
2431(-3)
3845(-3)
0407(-3)
5981(-5)
0390<-4)
8477(-3)
0258(-3)
l972(-3)
-48313 07707 35442 6384(-]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
29265
-97132
75770
-98389
25876
59832
13166
-24064
77451
-30586
12872
-50945
13656
50919
-14647
19383
-21497
22152
-21991
21378
REAL PART
61385
70701
69755
06768
62786
87990
18248
92480
94132
28761
89953
89650
84274
58155
36069
78645
79556
94753
29376
16118
39314
46799
64360
70942
24534
09323
82030
07684
43979
20954
32494
98019
78778
38987
46528
71833
93029
11399
09478
19285
2700C 0)
0875(-l)
6186(-1)
1158(-1)
6069( 0)
120K 0)
4871( 0)
946K-1)
2156(-2)
7520(-2)
3781(-2)
6893(-3)
1760(-3)
5174(-4)
8892(-3)
0517(-3)
7986(-3)
4375(-3)
8313(-3)
8994(-3)
A(I«J)
imaginary part
-35482 57378 86130 8098(
1
2
3
4
5
6
7
8
9
10
11
12
13
I*
15
16
17
18
19
20
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
27779
-79087
48368
-42560
49392
-78971
23615
59907
14269
-28183
98921
-43582
21468
-11127
57870
-28396
11387
-12902
-47894
84470
62647
00406
84886
86843
17446
76407
88831
49001
99028
79900
39234
83767
01620
48781
68075
93661
92204
73530
16016
36998
37844
55136
02638
08826
37191
27758
99095
50299
77211
41822
58009
94233
25615
45195
73325
37171
07397
68522
13769
40465
4672( 0)
1084(-1)
2446(-l)
9234<-l)
2956C-1)
4076(-l)
7260< 0)
0723< 0)
5707( 0)
195U-1)
9583<-2)
9l82(-2)
5729C-2)
5438(-2)
9320(-3)
1642<-3)
9519(-3)
6433(-4)
3077<-4)
5606(-4)
-63117 49480 66473 0072<

Ad,J)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
-27387
74719
-42768
34309
-34732
43759
-73791
22923
59927
14659
-29703
10710
-48665
24899
-13577
76091
-42376
22381
-10111
24069
REAL
88463
03786
39038
83517
54758
75765
81124
47662
22436
43083
64619
74686
74192
19657
43283
55873
80986
67999
02849
70634
. HART
17529
06319
35713
31896
86464
28787
26996
15085
78569
16707
9/777
47882
41964
61897
73372
06953
9b928
16676
5tt849
18338
6783( 0)
4275(-l)
8634(-l)
4005(-l)
1190(-1)
6129(-1)
231K-1)
9833( 0)
7937( 0)
611K 0)
2236(-l)
3423(-l)
8170(-2)
3064(-2)
5548(-2)
9666(-3)
9522(-3)
2456(-3)
661K-3)
0753(-4)
IMAGINARY PART
•79895 82978 18899 1547(-1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
27110
-71712
39119
-29398
27221
-30008
39948
-70005
22389
59941
14979
-30979
11409
-53060
27897
-15738
92286
-54887
32280
-18097
REAL PART
02491
09901
06387
69672
43614
47948
30651
73631
26030
23592
25743
23123
24252
22241
79828
01811
39537
63417
41641
06864
41454
57127
65692
99133
66467
71509
20875
37602
69543
97275
46065
27772
10401
71161
16286
90092
46855
81578
70520
15093
6967( 0)
3804(-l)
7109(-1)
5392(-l)
3805(-l)
0752(-l)
8354(-l)
806K-1)
8385( 0)
5432( 0)
5212( 0)
3789(-l)
4158(-1)
6516(-2)
3851(-2)
U66(-2)
2589(-3)
9580(-3)
3801(-3)
9509(-3)
Ad,J)
IMAGINARY PART
-98648 08339 62559 7000(-1)
1
2
3
4
5
6
7
8
9
10
U
12
13
14
15
16
17
18
19
20
-26905
69547
-36591
26199
-22771
23017
-26862
37199
-67117
21964
59951
15246
•32065
12012
•56897
30541
-17657
10677
-66149
41238
68752
69790
62996
33569
74545
60064
14284
82411
34272
53195
54678
59376
05347
28459
61077
06346
83454
60313
10089
63154
58684
17257
33681
60383
45199
43798
0*003
71240
53961
06525
8^156
59144
04327
6J132
51950
09870
17510
16258
41084
53426
1583( 0)
6904(-l)
774K-1)
3961(-1)
871M-1)
5420(-l)
3817(-1)
8066(-l>
5631(-1)
9933 ( 0)
3721 ( 0)
4016( 0)
5833(-l)
7200(-l)
62l6(-2)
6126(-2)
718K-2)
0132(-2)
1465(-3)
9l24(-3)
-11937 42556 49745 4643< 0)
1
2
3
4
5
6
7
8
9
10
11
12
15
16
12
12
12
12
12
12
12
12
12
12
12
12
13 12
14 12
12
12
17 12
18 12
19 12
20 12
26750
•67934
34760
•23980
19879
•18897
20256
•24621
35124
-64840
21618
59959
15473
-33000
12538
•60277
32888
•19375
11981
•76340
97920
82135
60321
50436
91412
21932
06180
05453
74987
99384
72256
35797
37919
54837
22003
89611
96143
27587
82770
71080
00055
98132
55454
75005
17023
82097
54121
01437
65587
39726
54721
69134
47364
01825
49122
83503
20360
87013
16747
24694
313K 0)
1335(-1)
1588(-1)
4359(-l)
1342(-1)
5780<-l>
7327(-l)
3779(-l)
8067(-l)
8295(-l)
2601( 0)
5215C 0)
8071( 0)
1419(-1)
2858(-1)
2701(-2)
124K-2)
5903(-2)
7429(-2)
0255(-3)
-14207 43465 42358 3984( 0)

A(ItJ)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
13
-26630
66699
-33387
22369
-17876
16228
-16381
18310
-22945
33502
-63000
21331
59965
15668
-33814
13000
-63278
34988
-20920
13162
REAL PART
99840
09134
28239
30909
06044
03439
61717
63814
46342
83158
64176
68708
41753
19753
93268
97641
4800b
67648
99748
18142
28339
5U717
75980
56658
6/242
38618
13501
40123
45485
9*305
3^102
53304
41129
49783
53454
06944
36816
88651
55184
16487
3860 (
9260(-
2818<-
1249(-
9562(-
8984(-
2150(-
1232<-
0674(-
7060 (•
9663<«
8630 (
3030 (
8974 (
1130(-
2034(«
430K-
6690(-
5551 <«
1097(«
0)
•1)
•1)
•1)
•1)
•1)
•1)
•1)
•1)
•1)
•1)
U)
0)
0)
•1)
•1)
•2)
•2)
•2)
•2)
IMAGINARY PART
•16674 83597 99385 0708( (
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
-26459
64956
-31493
20220
-15325
13042
-12155
12257
-13337
15759
-20609
31131
-60207
20882
59974
15985
•35163
13777
•68371
38586
63169
62406
44772
03196
05429
45588
07183
11859
08188
31231
21187
31225
16065
69032
07659
61834
69825
67883
39895
73240
63434
08692
1U044
17986
34983
63532
22396
29700
03086
49126
51257
1U910
54090
53793
86826
54394
22904
24117
65148
75661
2194( 0)
1692<-1>
8608(-l)
679K-1)
9687(-l)
8817(-1)
2784(-l)
7322(-D
B769(-l)
7681(-1)
6580(-l)
0342(-l)
9767(-l)
7l64( 0)
7517( 0)
3405( 0)
8729C-1)
6605(-l)
9496<-2>
5996(-2)
-22201 81487 50066 7572 ( (
Ad,J)
1 ]
2 ]
3 1
4 ]
5 ]
6 ]
7 )
8 ]
9 ]
10 ]
11 ]
12 ]
13 ]
14 ]
15 ]
16 ]
it ]
18 ]
19 ]
20 ]
14 26536
L4 -65730
L4 32328
L4 -21157
L4 16420
14 -14381
14 13880
14 -14626
L4 16869
L4 -21646
L4 32200
14 -61481
L4 21089
L4 59970
L4 15837
L4 -34530
14 13411
14 -65960
L4 36877
L4 -22319
REAl
05826
47861
43382
39462
70431
96073
78124
96096
06348
04025
31084
89073
61232
21421
35812
30204
31480
16290
81398
74445
. PART
19905
45687
28404
89718
26225
28623
72805
85987
91390
89291
49280
42851
00654
79012
45541
40042
87635
13195
18459
74907
4718( 0)
1033(-1)
5219(-1)
151K-1)
4715(-1)
5332(-l)
3808(-l)
234K-1)
8005(-l)
3342(-l)
2620(-1)
1620(-D
9835( 0)
2986( 0)
5726( 0)
0189(-1)
6363(-l)
6339(-2)
7492(-2)
4221(-2)
IMAGINARY PART
-19339 62944 89502 9068( 0)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1?
is
19
20
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
26397 19359
-64328 24861
30822 52689
-19478 40675
14476 48776
-12034 27882
10904 80294
-10626 04016
11072 28884
-12350 82198
14879 22994
-19762 82452
30238 20067
-59121 96729
20703 78065
59977 23266
16U6 63084
-35728 45179
14106 78809
-70551 28284
57670
25826
63724
71163
22947
86507
07858
62439
93910
40005
41993
23313
75587
33220
10925
48235
47718
58760
04302
26435
2118C 0)
8356(-l)
3748(-l)
1775(-1)
0968(-l)
1297(-1)
8486(-1)
1079(-1)
4319(-1)
0402(-D
2365<-l>
5700<-l>
7972<-l)
8633(-1)
2930< 0)
3210< 0)
6203< 0)
2615(-1)
ooio(-i)
5898(-2)
-25261 39244 43721 77l2< 0)

Ad,J)
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
n
n
ii
ii
ii
ii
r
ii
r
r
ii
r
ii
r
r
r
r
r
r
r
' -26345
r 63810
r -30274
r 18880
r -13804
r 11253
r -99646
r 94464
r -95174
r 10171
r -11573
r 14164
r -19058
r 29480
r -58186
f 20547
r 59979
r 16233
r -36235
r 14404
REAL HART
52158
83093
79653
51629
02047
26985
24419
52138
32772
94066
14667
49493
88994
85777
94597
55650
84409
23260
13923
05915
41531
58358
24670
06646
65275
4*330
80838
94546
55445
66234
94386
26291
22436
40152
11015
46072
33227
36427
95 794
49128
7535 (
3832(-
0764(-
7285(-
1207(-
2345<-
7756(-
0320(-
7188(-
1356(-
506K-
6803(-
9526(-
8778(-
2245(-
0062 (
5391 (
4979(
3916(-
2940(-
0)
•1)
•1)
•1)
-1)
-1)
-2)
-2)
-2)
•1)
•1)
-1)
•1)
-1)
-1)
0)
0)
0)
-1)
-1)
IMAGINARY PART
-28518 36159 82532 5012( <
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
-26265
63016
-29441
17984
-12815
10133
-86593
78750
-75533
76046
-80258
88976
-10426
13074
-17955
28265
-56657
20287
59983
16431
70798
21533
87121
20283
17464
35841
25889
42036
80390
61397
62742
63001
42700
61038
34067
94957
80929
81957
87932
78015
45666
93030
4O580
01209
66755
70788
31173
1/572
16351
6J593
21539
12336
48588
23184
78723
1*051
6^855
9^436
7/740
94734
8854 (
1666(-
2193(-
2113(-
0)
•1)
•1)
•1)
8670(-l)
5265(-
0862(-
7844(-
1038(-
6737(-
4974(-
8740(-
5143(-
9670(-
0015(-
8894<-
1750(-
5997 (
4785 (
1919(
•1)
•2)
•2)
•2)
•2)
•2)
•2)
•1)
•1)
•1)
•1)
•1)
0)
0)
0)
-35624 47726 72653 1982( (
Ad,J)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
18
26302
-63379
29821
-18390
13260
-10633
92365
-85605
83951
-86797
94657
-10944
13572
-18464
28830
-57372
20409
59982
16337
-36692
REAL PART
27282
55407
49692
79870
89620
96979
90936
20596
09674
12424
69398
66028
65945
26725
50311
92440
95229
03083
67466
29894
64331
79829
25227
84239
12454
58901
77997
80223
89331
05474
95749
73504
98963
38757
35678
60814
03561
87279
99213
87648
8176(
0253(-
8328<-
7213(-
8674(-
5245(-
589H-
4650(-
9607(-
1659<-
092K-
1618(-
8567(-
3242(-
2913<-
3606(-
5921 (
5105(
9616(
0103(-
0)
•1)
-1)
-1)
-1)
-1)
-2)
-2)
-2)
-2)
•2)
-1)
-1)
•1)
-1)
•1)
0)
0)
0)
-1)
IMAGINARY PART
-31972 72308 17079 5441( 0)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
26234
-62707
29120
-17642
12444
-97220
81925
-73317
69027
-67995
69904
-75021
84311
-99919
12649
-17514
27771
-56024
20178
59985
51724 64847 5647( 0)
18947 05295 5627(-l)
62196 99233 7704<-l>
62723 73646 4977(-l)
36891 28383 9941(-1)
99523 06002 3785<-2)
87876 69241 4284(-2)
66129 93702 2924<-2)
69091 53908 4911(-2)
90366 42685 5326(-2)
54625 33831 5964(-2)
08055 63256 1445(-2)
18563 74889 6122C-2)
05535 57127 7l42<-2)
73520 30128 2406(-l)
63767 29339 3612(-1)
26268 48369 8368(-l)
61285 88914 8712<-1)
70470 88146 2097( 0)
45661 56864 1663( 0)
-39473 62303 73382 5684( 0)

EIGENVALUES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0*1900388bJ
0*l834b277b
0*59293030b
0.b3b220447
0.481285813
0.<*13414b9u
0* 7 77 7b408:>
0*3477b0817
0. 7b393287o
0. 78bB7bbbJ
0*305b0937*
0*8038l4bbb
0* 7339b97<fJ
0««1277b4bJ
0*690bbl40<f
0*203803b5n
0*84120403:*
0*839620250
0*b4203bbb<*
0*bbb4575l0
CONDITION NUMbEKS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
HEAL PARI
-0*032B7b3bb
-0*01354b439
-0*002773871
0*0045063bb
-0«002554UbJ
-0*0032904b7
0«0185359o<f
0*00526280*:
0.003007399
-0*005432201
-0*004813871
0*050142980
-0*00112150*
0*017361091
-0*00125614*
0*044986778
1*22669722.*
-0.079497763
-0*00043194*
0*451106514
PEAL PANT
3975896*46 8207230665 1
3718224009 b583109292 2
61828b2b74 7122bb4l54 7
632971597b 03067b4219 8
77706b7b76 1517590063 b
1693897078 b280776578 0
9441888714 b733245007 7
9bb2195079 9977073364 7
7907607449 bl088b0248 1
7l4953bbl7 b0 744b33b9 b
1347431406 2009523722 4
3346488051 3144306437 6
5409688bi9 5522b207b2 1
0935692249 1572885856 4
94b77073t*8 0128713971 7
b24005409b 08887b4bb4 0
505962062b 9b79400580 7
990b8574b0 1997b23006 9
7350083bb2 b943022874 0
3803274381 6689632229 9
IMAG* PART
-0.088548428
-0.034630907
-0*00*936017
-0*000781866
0*004554858
-0*00*946580
-0*014170278
-0*000995753
-0*000360438
-0.001265974
0*004723745
-0*220685641
0*00l78b629
0*009402279
-0*002492914
0*054156178
0*03987b322
-0*002272420
0*003393100
-0*059596278
IMAGINARY PART
-0*215008457
-0*176568538
-0.140888947
-0.141263130
-0*141103339
-0.141327999
-0*186925814
-0.141957761
-0.138787096
-0.113440005
-0.124485416
-0.247603569
-0.140217031
-0*086714318
-0*140154726
0.003326547
-0.323979871
-0.060060466
-0.141049064
-0.033404574
76020b4027
5870739008
35331bb860
4984021724
8793303898
0715091091
3349321476
5173601794
5640070089
3966406666
3169993538
0718583082
5179807143
4069559104
6558511635
8966812518
2081584 755
7273840046
8312920122
4580333028
4025322840 0
8862618827 b
0691745139 4
9845212029 6
5118564508 6
6275814065 7
9116030292 1
6905493933 6
9986346194 8
4357187600 8
0872511197 1
641b959528 7
2318078465 7
7463400092 8
8514821558 5
1715340224 6
7403882355 7
8752892080 2
78612l7b58 9
8428515962 7

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132

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o
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oc ^WH^o^^^ino^^NNSOHO**
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a. flO^N-tflonwinnoiAHoo^s^oo-t
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133

CHAPTER VII
TEST MATRICES: EIGENVALUES AND EIGENVECTORS
OF TRIDIAGONAL MATRICES
Example 7,1
Let A =
tti
an
a
15
>16
a,
16
where the a, and .p are given by
a.
3 + 2i
1 - i
3 - 4i
2 + 3i
5 + i
1 + 2i
5 + 21
2 + i
1 - 2i
1-41
2 + i
1 - 5i
3 + i
2 + 4i
4 + 3i
1 - 5i
4
2
3
3
2
2
1
-2
3
-1
4
1
2
5
-3
h
- i
+ 4i
+ i
- 2i
- 2i
+ 3i
+ 3i
+ 2i
+ 3i
+ 5i
+ 3i
- 61
+ i
i
- 41
134

Eigenvalues:
Eigensystems—Tridiagonal Matrices
135
2.06853152
2.40341933
2.72491267
2.45640400
2.2774q066
-
+
-
+
+_
0.812811959 +
-1.38565721
-2.72480368
Reference: [65]
Example 7.2
10
1
.._
+
•
1
9
1
2.054430451
2.081055121
2.37837845i
0.63193686H
1.448268501
1.335511351
1.3875605H
0.657064546i
1
8 1
1
3
1
3
2
4
5
5
.57598142 -
.28048252 +
.19252750 -
.55339888 +
.45560768 -
.89673115 +
.65716067 +
.77408994 +
3.83032770i
3.275661631
5.443997521
1.264656311
4.692904961
3.622108561
1.632000821
2.839335911
1 1
10 1
1 1
_
igenvalues:
10.74619 42
10.74619 42
9.21067 86
9.21067 86
8.03894 11
8.03894 11
7.00395 22
7.00395 18
6.00023 40
6.00021 75
5.00024 44
4.99978 25
4.00435 40
3.99604 82
3.04309 93
2.96105 89
2.13020 92
1.78932 14
0.94753 44
0.25380 58
-1.12544 15
1
8
1
1
9
1
1
10
Reference: [62, p. 309]-

136 Matrices for Testing Computational Algorithms
Example 7.3
10
1
1
9 1
18 1
Eigenvalues:
1 1
1 0
1
±10,74619 42
± 9.21067 86
± 8.03894 11
± 7.00395 20
± 6.00022 57
± 5.00000 82
+ 4.00000 02
± 3.00000 00
± 2.00000 00
± 1.00000 00
0.00000 00
1
■1 1
-8 1
1-9 1
1 -10
Reference: [62, p. 309].

Example 7.4
Eigensystems—Tridiagonal Matrices 137
a b
b a b
b a b
b a b
b a
, n x n.
Eigenvalues:
Xk =
a + 2b cos ^7 , k = 1, 2,
rri-1
n.
Eigenvectors:
,<k> _
4k)
,00
.(k)
, where x
J
■('
'£) - &
j = 1, 2, .... n; k = 1, 2,
n.
Reference: [18, pp. 20, 25]
Example 7.5
(a-b) b
b a
b
b
a
b
b
a
b
b
a
n x n.

138 Matrices for Testing Computational Algorithms
Eigenvalues:
\ -a + 2b cos i+T
y Jv~~JLj C) ■ ■ ■ j II «
Reference: [18, pp. 27-28].
Example 7.6
(a-b) b
b a b
b a b
b a b
b (a+b)
, n x n.
Eigenvalues:
X. = a + 2b cos (2k~l)rt , k = 1, 2, ..., n.
K £31
Reference: [18, pp. 27-28]
Example 7.7
(a+b) b
b a
b
b
a
b
b
a b
b (a+b)
n x n.
Eigenvalues:
Xfc = a + 2b cos — (k-1), k = 1, 2, . . ., n.
Reference: [18, p. 29].

Example 7.8
Eigensystems—Tridiagonal Matrices 139
a 1
1 b
1
1
a
a
1
1
b
, 2n x 2n,
Eigenvalues:
a + b +
Xk =
jVb)
2+16cos2 krt
2n+
a"
, k - 1, 2,
n.
Reference: [18, pp. 31-32]
Example 7.9
a
1
1
b 1
1 a
b
1
1
a
2n+l x 2n+l
Eigenvalues:
a + b
X, =
+ [(a-b)2 + 16 cos2 -^
2n+2.
J
Ix ~" ^ , ^—9 * * " > 11 •
X2n+1 =a
Reference: [18, p, 33].

140 Matrices for Testing Computational Algorithms
Example 7.10
Let A 4, be the n+1 x nfl matrix
rri-1
yx o
y2 o
y . 0 x
n-1 n
*n °
with
x-y, = k(n-fefl), k = 1, 2, ..., n.
Eigenvalues:
If n is an odd integer, the eigenvalues of A . 1 are
±n, ±(n-2), ..., ±1-
If n is an even integer, the eigenvalues of A 1 are
±n, ±(n-2), .,., ±2, 0.
Examples:
yk = n - k + 1, k - 1, 2, ...,
B. x, = i[k(n-k+-l)]% k * 1, 2, ..., n.
yfc = -i[k(n-kfl)]*, k = 1, 2, ..., n.
Inverse: Example 3-19
Reference: [12],

Eigensystems—Tridiagonal Matrices 141
Example 7.11
Let A = [a ,] be the n+1 x n+1 matrix given by
A =
n
n
0
0
n + s
-(3n+s-2)
2(n-l)
0
0
2(n+s-l)
-(5n+2s-8)
3(n-2)
0
0
3(n+s-2)
-(7n+3s-18)
where s is an arbitrary parameter. In general,
aii = -t<2l+l)n + is - 2i ]
ai,i+l * (i+l)(n+s-i)
ai,i-l = i^""1*1)
a±. = 0, if |i-j| > 1,
where i,j =0, 1, 2, ..., n.
Eigenvalues:
\j = -j(s+j+l), j - 0, I, 2, .... n.
Left Eigenvectors:
Let y ■*' be the left eigenvector of A corresponding to the eigenvalue
^-.» j = 0, 1, ..., n. Then the components of y^' are given by
rU) _
ft i^ (s) d) m
for i = 0, 1, 2, ..., n, and q ■ min(i,j).

142 Matrices for Testing Computational Algorithms
Right Eigenvectors:
Let x be the right eigenvector of A corresponding to the eigenvalue
^> j = 0, 1, . . . , n, Let r be an integer such that I ^ r ^ n. The components
of x^J' are
,.N /n+s-i\ (
x<j) = [ n-i j ^1
unless s = -r and j ^ r.
If 8 * -r and j ^ r, then
xa> = 1 U-i1) ly(
xi |_ (r+s) Jyi
(j) If i s n - r,
c(j) = f (t'l I yJ— if i > „
ci |_ (r+s) J (r+s) " x > n
Reference: [16], [60, pp. 156-157].

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8 (1959), 10-12.
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8. Brenner, J. L., and G.W. Reitweisner. "Remark on Determination of
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the Characteristic Roots and Vectors of a Real Symmetric Matrix," Proc. IEE,
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ACM, 6 (1963), 310-
12. Clement, Paul A. "A Class of Triple-diagonal Matrices for Test Purposes,"
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Eigenvalues and Eigenvectors," Jour, SIAM, 10 (1962), 74-88,
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144 Matrices for Testing Computational Algorithms
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6 (1958), 378-392.
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SYMBOL TABLE
an n x n matrix
an n x n matrix whose i,j element is a..
the i,j submatrix of a partitioned matrix
the complex conjugate of A
the complex conjugate transpose of A
the transpose of A
the inverse of A
the norm of A
(i) the kfc matrix in a sequence
(ii) the k Kronecker power of A
the Kronecker product of A and B
the determinant of A
general condition number of A
Turing1s M-condition number
Turing1s N-condition number
J (i) von Neumann and Goldstinefs condition number
(ii) a polynomial in A
the identity matrix of order n
the identity matrix of order m
an n-dimensional column vector
148

Symbol Table 149
T
x the corresponding row vector (x transpose)
ij
x the complex conjugate transpose of x
(x,y)
H
y x
the scalar product of x and y
H H
xy a square matrix (do not confuse with y x)
x the j vector in a sequence
x. the i1" component of x^
an eigenvalue
X. the i eigenvalue
P(X) the characteristic polynomial
Is.l the condition number of X.
l " i
(k,n) the greatest common divisor of k and n
(k,n) =1 k and n are relatively prime
/ \ r
exp(r) e
a) cube root of unity
§.. the Kronecker delta = 1, if i = j, 0, otherwise
J , n x k matrix, each element is 1
nk
f the n-dimensional vector, each component is 1

150 Symbol Table
e column vector of dimension n whose components are 8..,
J = 1,2,...,n
g. f - ne i = l,2,...,n-l
l n n '
a = b (mod m) a - b is divisible by m
a = b a is approximately equal to b
a ~ b a is asymptotic to b
n n n J r n
i")
equivalence relation
binomial coefficient
a |b a divides b
|a| absolute value of a
i \Tl

INDEX
Bellman, 7
Binomial coefficients, 32, 43
Block-diagonal matrix, 6, 13, 21, 22, 24-26
Brenner, 12, 23
Cauchy's matrix, 54
Characteristic
equation, 6, 16, 104
polynomial, 57, 74, 90, 105
Circulants, 22, 23
Cline, 17
Combinatorial matrix, 53
Complete set (of eigenvectors), 18
Complex matrix, 17
Composite matrix, 6, 7, 12
Compound matrix, 6
Conjugate transpose, 1
Condition number
eigenvalue problem, 4, 22
inversion problem, 2, 3
Defective (matrix), 82, 86, 91, 106, 108, 110
Determinant, 11, 14, 23, 38
example, 50
Diagonal matrix, 15
Dolph-Lewis matrix, 122-133
Dominant eigenvalue, 79, 80
Elementary divisors, 23, 108, 110
151

152 Index
Elimination method, 89
Equivalence
classes, 25, 27
relation, 25
Forsythe, 28, 92, 103
Friedman, 7
Givens1 method, 1, 64
Goldstine (and von Neumann) condition number, 2
Hadamard matrices, 42
Hermitian (matrices), 1, 18, 21, 118
Hilbert matrices, 54
determinants, 38
eigenvalues, 66-73
inverses, 33-37
Householder1s method, 1, 58, 59
Ill-conditioned matrices, 2
Invariant (sub)space, 12, 56, 62, 82, 86, 91, 106, 109, 110, 115, 117
Jordan form, 28, 109, 113
K(A), general condition number (definition), 3
Kronecker
powers, 8
product, 7, 8, 9
Lanczos1 method, 2, 61, 84
Left eigenvectors, 1, 4, 6, 21, 23, 81-86, 88, 90, 93, 103, 122, 141
Linearly independent set (of eigenvectors), 18
M(A), Turing1s M-condition number, 3

Index 153
Mahler matrix, 23
Marcus, 7
Matrix polynomial, 5
Matrix power, 11.
Multiple eigenvalue, 12, 14, 17, 19, 27, 56, 62, 82, 86, 91, 106, 108, 110, 115, 117
N(A), Turing1s N-condition number, 3
Newberry's method, 15
Nonsingular matrix, 2, 6, 8, 24
Nonsymmetric matrix, 2, 21, 22
Normalized (eigenvectors), 4
Nth roots of unity, 26, 27, 83
Ortega, 19
Orthogonal
matrix, 20, 41
vectors, 56, 62, 115
Orthonormal, 17-19
P(A), von Neumann and Goldstine's condition number, 2
Pascal's matrix, 32
Pei's matrix, 18
Polynomial (matrix), 5
Principal vectors, 28, 109-113
Rosser's matrix, 61
|s.| (condition number for X.), 4
Similarity transformation, 19, 20, 117
Submatrix, 6, 15
Symmetric matrix, 1, 3, 20, 23, 34

154 Index
Subspace (invariant), 12, 56, 62, 82, 86, 91, 106, 109, 110, 115, 117
Turing, condition number, 3
Tridiagonal form, 1, 2, 58, 59, 61, 64, 84, 89
Vandermonde1s matrix, 27, 28, 54
Varahfs program, 28
von Neumann (and Goldstine) condition number, 2
Well-conditioned matrices, 2
Wilkinson, 122