Matrix AnalysisLinear 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. 
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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 
543  
Notation  547 
549  
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Common terms and phrases
algebraic characteristic polynomial coefficients commute complex numbers conclude condition number congruence Consider convergence convex Corollary corresponding defined denote diagonal matrix diagonalizable direct sum doubly stochastic eigen eigenvalues eigenvector equal equations Euclidean example Exercise factorization finite function GerSgorin given matrix hence Hermitian matrix Hint identity inequality inner product integer inverse irreducible Jordan blocks Jordan canonical form Lemma Let A,BeMn Let AeMn linearly independent main diagonal entries matrix AeMn matrix norm minimal polynomial modulus multiplicity node nonnegative matrix nonsingular matrix nonzero vector norm on Mn normal matrices orthonormal pA(t permutation matrix positive semidefinite Problem Proof rank real numbers real orthogonal real symmetric result row and column scalar sequence Show similar singular value decomposition spectral norm suppose symmetric matrix Theorem tion UeMn unit ball unitarily equivalent unitarily invariant norm unitary matrix upper triangular matrix vector norm vector space x*Ax zero