Page 11 - Matrices theory and applications

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The Spectrum and the Diagonal of Hermitian Matrices .
51
3.4
3.5
Exercises . ... .. .. ... .. .. ... .. .. ... ..
55
4 Norms
A Brief Review .. .. ... .. .. ... .. .. ... ..
4.1
61
Householder’s Theorem ... .. .. ... .. .. ... ..
4.2
66
An Interpolation Inequality
4.3 Contents . .. .. . .. .. .. . .. . 61
67
4.4 A Lemma about Banach Algebras . . . . . . . . . . . . . 70
4.5 The Gershgorin Domain .. .. .. ... .. .. ... .. 71
4.6 Exercises . ... .. .. ... .. .. ... .. .. ... .. 73
5 Nonnegative Matrices 80
5.1 Nonnegative Vectors and Matrices . ... .. .. ... .. 80
5.2 The Perron–Frobenius Theorem: Weak Form . . . . . . . 81
5.3 The Perron–Frobenius Theorem: Strong Form . . . . . . 82
5.4 Cyclic Matrices .. .. ... .. .. ... .. .. ... .. 85
5.5 Stochastic Matrices .. ... .. .. ... .. .. ... .. 87
5.6 Exercises . ... .. .. ... .. .. ... .. .. ... .. 91
6 Matrices with Entries in a Principal Ideal Domain;
Jordan Reduction 97
6.1 Rings, Principal Ideal Domains .. ... .. .. ... .. 97
6.2 Invariant Factors of a Matrix . . . . . . . . . . . . . . . . 101
6.3 Similarity Invariants and Jordan Reduction . . . . . . . 104
6.4 Exercises . ... .. .. ... .. .. ... .. .. ... .. 111
7 Exponential of a Matrix, Polar Decomposition, and
Classical Groups 114
7.1 The Polar Decomposition .. .. .. ... .. .. ... .. 114
7.2 Exponential of a Matrix . . . . . . . . . . . . . . . . . . 116
7.3 Structure of Classical Groups . . . . . . . . . . . . . . . 120
7.4 The Groups U(p, q) . .. . .. .. .. . .. .. .. . .. . 122
7.5 The Orthogonal Groups O(p, q) .. ... .. .. ... .. 123
7.6 The Symplectic Group Sp .. .. ... .. .. ... .. 127
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7.7 Singular Value Decomposition . . . . . . . . . . . . . . . 128
7.8 Exercises . ... .. .. ... .. .. ... .. .. ... .. 130
8 Matrix Factorizations 136
8.1 The LU Factorization .. . .. .. .. . .. .. .. . .. . 137
8.2 Choleski Factorization ... .. .. ... .. .. ... .. 142
8.3 The QR Factorization . ... .. .. ... .. .. ... .. 143
8.4 The Moore–Penrose Generalized Inverse . . . . . . . . . 145
8.5 Exercises . ... .. .. ... .. .. ... .. .. ... .. 147

9 Iterative Methods for Linear Problems 149