Matrix Analysis

Front Cover
Cambridge University Press, Feb 23, 1990 - Mathematics - 561 pages
7 Reviews
Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research. In this book the authors present classical and recent results of matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematical science, but the necessary material has appeared only sporadically in the literature and in university curricula. As interest in applied mathematics has grown, the need for a text and reference offering a broad selection of topics in matrix theory has become apparent, and this book meets that need. This volume reflects two concurrent views of matrix analysis. First, it encompasses topics in linear algebra that have arisen out of the needs of mathematical analysis. Second, it is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis. Both views are reflected in its choice and treatment of topics.
 

What people are saying - Write a review

User ratings

5 stars
4
4 stars
3
3 stars
0
2 stars
0
1 star
0

User Review - Flag as inappropriate

This is by far my favorite textbook. Chapter 0 (the preliminaries) is boring, but includes some good information that may have been forgotten (e.g. 0.4.5 (c), probably the most useful rank inequality). The rest of the book is gold. The proofs are concise and well written. The problems are challenging and appropriate (if the problem is too hard, hints are usually given). The book is dense, so some may consider that a minus.  

Contents

Review and miscellanea
1
02 Matrices
4
03 Determinants
7
04 Rank
12
05 Nonsingularity
14
07 Partitioned matrices
17
08 Determinants again
19
09 Special types of matrices
23
53 Algebraic properties of vector norms
268
54 Analytic properties of vector norms
269
55 Geometric properties of vector norms
281
56 Matrix norms
290
57 Vector norms on matrices
320
58 Errors in inverses and solutions of linear systems
335
Location and perturbation of eigenvalues
343
61 Gerśgorin discs
344

010 Change of basis
30
Eigenvalues eigenvectors and similarity
33
11 The eigenvalueeigenvector equation
34
12 The characteristic polynomial
38
13 Similarity
44
14 Eigenvectors
57
Unitary equivalence and normal matrices
65
21 Unitary matrices
66
22 Unitary equivalence
72
23 Schurs unitary triangularization theorem
79
24 Some implications of Schurs theorem
85
25 Normal matrices
100
26 OR factorization and algorithm
112
Canonical forms
119
a proof
121
some observations and applications
129
the minimal polynomial
142
34 Other canonical forms and factorizations
150
35 Triangular factorizations
158
Hermitian and symmetric matrices
167
41 Definitions properties and characterizations of Hermitian matrices
169
42 Variational characterizations of eigenvalues of Hermitian matrices
176
43 Some applications of the variational characterizations
181
44 Complex symmetric matrices
201
45 Congruence and simultaneous diagonalization of Hermitian and symmetric matrices
218
46 Consimilarity and condiagonalization
244
Norms for vectors and matrices
257
51 Defining properties of vector norms and inner products
259
52 Examples of vector norms
264
62 Gerśgorin discs a closer look
353
63 Perturbation theorems
364
64 Other inclusion regions
378
Positive definite matrices
391
71 Definitions and properties
396
72 Characterizations
402
73 The polar form and the singular value decomposition
411
74 Examples and applications of the singular value decomposition
427
75 The Schur product theorem
455
products and simultaneous diagonalization
464
77 The positive semidefinite ordering
469
78 Inequalities for positive definite matrices
476
Nonnegative matrices
487
81 Nonnegative matrices inequalities and generalities
490
82 Positive matrices
495
83 Nonnegative matrices
503
84 Irreducible nonnegative matrices
507
85 Primitive matrices
515
86 A general limit theorem
524
87 Stochastic and doubly stochastic matrices
526
Complex numbers
531
Convex sets and functions
533
The fundamental theorem of algebra
537
Continuous dependence of the zeroes of a polynomial on its coefficients
539
Weierstrasss theorem
541
References
543
Notation
547
Index
549
Copyright

Other editions - View all

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information