Contents
Preface
page IX
Chapter 0 Review and miscellanea
0.0 Introduction
0.1
Vector spaces
0.2
Matrices
Determinants
0.3
Rank
0.4
Nonsingularity
0.5
0.6
The usual inner product
0.7
Partitioned matrices
Determinants again
0.8
0.9
Special types of matrices
0.10 Change of basis
1
1
1
4
7
12
14
14
17
19
23
30
Chapter 1 Eigenvalues, eigenvectors, and similarity
1.0
Introduction
The eigenvalue-eigenvector equation
1.1
1.2
The characteristic polynomial
1.3
Similarity
1.4
Eigenvectors
33
33
34
38
44
57
Chapter 2 Unitary equivalence and normal matrices
2.0 Introduction
2.1 Unitary matrices
65
65
66
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vi
Contents
2.2
2.3
2.4
2.5
2.6
Unitary equivalence
Schur's unitary triangularization theorem
Some implications of Schur's theorem
Normal matrices
QR factorization and algorithm
72
79
85
100
112
Chapter 3 Canonical forms
3.0
Introduction
3.1
The Jordan canonical form: a proof
3.2
The Jordan canonical form: some observations
and applications
3.3
Polynomials and matrices: the minimal
polynomial
3.4
Other canonical forms and factorizations
3.5
Triangular factorizations
119
119
121
Chapter 4 Hermitian and symmetric matrices
4.0
Introduction
4.1
Definitions, properties, and characterizations of
Hermitian matrices
4.2
Variational characterizations of eigenvalues of
Hermitian matrices
4.3
Some applications of the variational
characterizations
4.4
Complex symmetric matrices
4.5
Congruence and simultaneous diagonalization
of Hermitian and symmetric matrices
4.6
Consimilarity and condiagonalization
167
167
Chapter 5 Norms for vectors and matrices
5.0
Introduction
5.1
Defining properties of vector norms and inner
products
5.2
Examples of vector norms
5.3
Algebraic properties of vector norms
5.4
Analytic properties of vector norms
5.5
Geometric properties of vector norms
5.6
Matrix norms
5.7
Vector norms on matrices
5.8
Errors in inverses and solutions of linear
systems
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129
142
150
158
169
176
181
201
218
244
257
257
259
264
268
269
281
290
320
335
Contents
vii
Chapter 6 Location and perturbation of eigenvalues
6.0
Introduction
6.1
Gersgorin discs
6.2
Gersgorin discs - a closer look
6.3
Perturbation theorems
6.4
Other inclusion regions
343
343
344
353
364
378
Chapter 7 Positive definite matrices
7.0
Introduction
7.1
Definitions and properties
7.2
Characterizations
7.3
The polar form and the singular value
decomposition
7.4
Examples and applications of the singular value
decomposition
The Schur product theorem
7.5
Congruence: products and simultaneous
7.6
diagonalization
7.7
The positive semidefinite ordering
Inequalities for positive definite matrices
7.8
391
391
396
402
Chapter 8 Nonnegative matrices
Introduction
8.0
8.1
Nonnegative matrices - inequalities and
generalities
8.2
Positive matrices
Nonnegative matrices
8.3
8.4
Irreducible nonnegative matrices
Primitive matrices
8.5
A general limit theorem
8.6
8.7
Stochastic and doubly stochastic matrices
Appendices
A
B
C
D
E
Complex numbers
Convex sets and functions
The fundamental theorem of algebra
Continuous dependence of the zeroes of a
polynomial on its coefficients
Weierstrass's theorem
References
Notation
Index
411
427
455
464
469
476
487
487
490
495
503
507
515
524
526
531
533
537
539
541
543
547
549
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