| Preface to the Second Edition |
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vii | |
| Preface |
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ix | |
| Frequently Used Notation and Terminology |
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xv | |
| Frequently Used Theorems |
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xvii | |
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1 Elementary Linear Algebra Review |
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1 | (34) |
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1 | (7) |
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1.2 Matrices and Determinants |
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8 | (9) |
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1.3 Linear Transformations and Eigenvalues |
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17 | (10) |
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27 | (8) |
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2 Partitioned Matrices, Rank, and Eigenvalues |
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35 | (38) |
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2.1 Elementary Operations of Partitioned Matrices |
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35 | (7) |
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2.2 The Determinant and Inverse of Partitioned Matrices |
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42 | (9) |
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2.3 The Rank of Product and Sum |
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51 | (6) |
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2.4 The Eigenvalues of AB and BA |
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57 | (5) |
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2.5 The Continuity Argument and Matrix Functions |
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62 | (5) |
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2.6 Localization of Eigenvalues: The Gersgorin Theorem |
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67 | (6) |
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3 Matrix Polynomials and Canonical Forms |
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73 | (34) |
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73 | (6) |
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3.2 Matrix Decompositions |
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79 | (8) |
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3.3 Annihilating Polynomials of Matrices |
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87 | (6) |
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3.4 Jordan Canonical Forms |
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93 | (9) |
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3.5 The Matrices AT, A, A*, AT A, A*A, and AA |
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102 | (5) |
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4 Numerical Ranges, Matrix Norms, and Special Operations |
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107 | (18) |
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4.1 Numerical Range and Radius |
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107 | (6) |
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113 | (4) |
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4.3 The Kronecker and Hadamard Products |
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117 | (5) |
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122 | (3) |
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5 Special Types of Matrices |
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125 | (46) |
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5.1 Idempotence, Nilpotence, Involution, and Projections |
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125 | (8) |
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133 | (5) |
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138 | (5) |
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143 | (7) |
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150 | (5) |
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5.6 Permutation and Doubly Stochastic Matrices |
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155 | (9) |
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164 | (7) |
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6 Unitary Matrices and Contractions |
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171 | (28) |
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6.1 Properties of Unitary Matrices |
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171 | (6) |
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6.2 Real Orthogonal Matrices |
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177 | (5) |
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6.3 Metric Space and Contractions |
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182 | (6) |
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6.4 Contractions and Unitary Matrices |
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188 | (4) |
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6.5 The Unitary Similarity of Real Matrices |
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192 | (3) |
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6.6 A Trace Inequality of Unitary Matrices |
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195 | (4) |
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7 Positive Semidefinite Matrices |
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199 | (54) |
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7.1 Positive Semidefinite Matrices |
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199 | (8) |
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7.2 A Pair of Positive Semidefinite Matrices |
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207 | (10) |
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7.3 Partitioned Positive Semidefinite Matrices |
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217 | (10) |
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7.4 Schur Complements and Determinant Inequalities |
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227 | (7) |
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7.5 The Kronecker and Hadamard Products of Positive Semidefinite Matrices |
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234 | (6) |
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7.6 Schur Complements and the Hadamard Product |
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240 | (5) |
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7.7 The Wielandt and Kantorovich Inequalities |
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245 | (8) |
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253 | (40) |
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8.1 Hermitian Matrices and Their Inertias |
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253 | (7) |
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8.2 The Product of Hermitian Matrices |
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260 | (6) |
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8.3 The Min-Max Theorem and Interlacing Theorem |
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266 | (8) |
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8.4 Eigenvalue and Singular Value Inequalities |
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274 | (7) |
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8.5 Eigenvalues of Hermitian matrices A, B, and A + B |
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281 | (6) |
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8.6 A Triangle Inequality for the Matrix (A*A)1/2 |
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287 | (6) |
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293 | (32) |
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9.1 Equivalent Conditions |
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293 | (13) |
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9.2 Normal Matrices with Zero and One Entries |
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306 | (6) |
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9.3 Normality and Cauchy-Schwarz-Type Inequality |
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312 | (7) |
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9.4 Normal Matrix Perturbation |
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319 | (6) |
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10 Majorization and Matrix Inequalities |
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325 | (54) |
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10.1 Basic Properties of Majorization |
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325 | (9) |
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10.2 Majorization and Stochastic Matrices |
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334 | (6) |
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10.3 Majorization and Convex Functions |
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340 | (9) |
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10.4 Majorization of Diagonal Entries, Eigenvalues, and Singular Values |
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349 | (7) |
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10.5 Majorization for Matrix Sum |
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356 | (7) |
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10.6 Majorization for Matrix Product |
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363 | (9) |
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10.7 Majorization and Unitarily Invariant Norms |
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372 | (7) |
| References |
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379 | (12) |
| Notation |
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391 | (4) |
| Index |
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395 | |