Introduction to Matrix Analysis

Title page......Page 1
Preface......Page 5
. Maximization of Functions of One Variable......Page 19
. Maximization of Functions of Two Variables......Page 20
. Algebraic Approach......Page 21
. Analytic Approach......Page 22
. Analytic Approach-II......Page 24
. A Simplifying Transformation......Page 25
. Definite and Indefinite Forms......Page 26
. Geometric Approach......Page 27
. Discussion......Page 28
. Vectors......Page 30
. Vector Addition......Page 31
. The Inner Product of Two Vectors......Page 32
. Orthogonality......Page 33
. Matrices......Page 34
. Matrix Multiplication-Vector by Matrix......Page 35
. Matrix Multiplication-Matrix by Matrix......Page 36
. Associativity......Page 38
. Invariant Vectors......Page 39
. Quadratic Forms as Inner Products......Page 40
. Symmetric Matrices......Page 41
. Hermitian Matrices......Page 42
. Unitary Matrices......Page 43
. The Solution of Linear Homogeneous Equations......Page 50
. Characteristic Roots and Vectors......Page 52
. Two Fundamental Properties of Symmetric Matrices......Page 53
. Reduction to Diagonal Form-Distinct Characteristic Roots......Page 55
. Reduction of Quadratic Forms to Canonical Form......Page 57
. Positive Definite Quadratic Forms and Matrices......Page 58
. Gram-Schmidt Orthogonalization......Page 62
. On the Positivity of the D_k......Page 65
. An Identity......Page 67
. The Diagonalization of General Symmetric Matrices-Two-dimensional......Page 68
. N-dimensional Case......Page 69
. Characteristic Vectors Associated with Multiple Characteristic Roots......Page 72
. The Cayley-Hamilton Theorem for Symmetric Matrices......Page 73
. Simultaneous Reduction to Diagonal Form......Page 74
. Simultaneous Reduction to Sum of Squares......Page 76
. Hermitian Matrices......Page 77
. Perturbation Theory-I......Page 78
. Perturbation Theory-II......Page 79
. Determinantal Criteria for Positive Definiteness......Page 90
. Representation as Sum of Squares......Page 92
. Constrained Variation and Finsler's Theorem......Page 93
. The Case k = 1......Page 95
. A Minimization Problem......Page 98
. Rectangular Arrays......Page 99
. Composite Matrices......Page 100
. The Result for General k......Page 102
. Functions of Symmetric Matrices......Page 107
. Uniqueness of Inverse......Page 108
. Square Roots......Page 110
. Parametric Representation......Page 111
. The Fundamental Scalar Functions......Page 112
. The Infinite Integral . . .......Page 114
. Relation between J(H) and |H|......Page 116
. The Rayleigh Quotient......Page 128
. Variational Description of Characteristic Roots......Page 129
. Geometrie Preliminary......Page 130
. The Courant-Fischer min-max Theorem......Page 131
. A Sturmian Separation Theorem......Page 133
. The Poincaré Separation Theorem......Page 134
. A Representation Theorem......Page 135
. Approximate Techniques......Page 136
. Integral Version......Page 141
. Hölder Inequality......Page 142
. Second Method......Page 143
. A Useful Inequality......Page 144
. Concavity of lambda_N......Page 145
. Additive Inequalities from Multiplicative......Page 146
. An Alternate Route......Page 147
. A Simpler Expression for lambda_N......Page 148
. Arithmetic-geometric Mean Inequality......Page 149
. Multiplicative Inequalities from Additive......Page 150
. A Problem of Minimum Deviation......Page 158
. Functional Equations......Page 159
. A More Complicated Example......Page 160
. Sturm-Liouville Problems......Page 161
. Functional Equations......Page 162
. Jacobi Matrices......Page 164
. Nonsymmetric Matrices......Page 165
. Complex A......Page 166
. Slightly Intertwined Systems......Page 167
. Simplifications-II......Page 168
. The Equation Ax = y......Page 169
. Quadratic Deviation......Page 170
. A Result of Stieltjes......Page 171
. Motivation......Page 177
. Vector-matrix Notation......Page 178
. Norms of Vectors and Matrices......Page 179
. Existence and Uniqueness of Solutions of Linear Systems......Page 181
. The Matrix Exponential......Page 183
. Functional Equations-I......Page 184
. Nonsingularity of Solution......Page 185
. Inhomogcneous Equation-Variable Coefficients......Page 187
. Perturbation Theory......Page 188
. Non-negativity of Solution......Page 190
. Polya's Fonctional Equation......Page 191
. The Equation dX/dt = AX + XB......Page 193
. Euler's Method......Page 201
. Nonsingularity of C......Page 202
. The Vandermonde Determinant......Page 204
. Diagonalization of a Matrix......Page 205
. Connection between Approaches......Page 206
. Multiple Characteristic Roots......Page 208
. Jordan Canonical Form......Page 209
. Multiple Characteristic Roots......Page 210
. Semidiagonal or Triangular Form-Schur's Theorem......Page 213
. Normal Matrices......Page 215
. An Approximation Theorem......Page 216
. Another Approximation Theorem......Page 217
. Alternate Proof of Hamilton-Cayley Theorem......Page 218
. Linear Equations with Periodic Coefficients......Page 219
. A Nonsingular Matrix Is an Exponential......Page 220
. An Alternate Proof......Page 222
. Some Interesting Transformations......Page 223
. Biorthogonality......Page 224
. The Laplace Transform......Page 226
. An Example......Page 227
. Discussion......Page 228
. Matrix Case......Page 229
. Powers of Characteristic Roots......Page 241
. Polynomials and Characteristic Equations......Page 243
. Symmetric Functions......Page 244
. Kronecker Products......Page 245
. Kronecker Powers-II......Page 246
. Kronecker Logarithm......Page 247
. Kronecker Sum-II......Page 248
. The Equation AX + XB = C......Page 249
. An Alternate Route......Page 250
. Circulants......Page 252
. A Necessary and Sufficient Condition for Stability......Page 258
. A Method of Lyapunov......Page 260
. Mean Square Deviation......Page 261
. Effective Tests for Stability......Page 262
. A Necessary and Sufficient Condition for Stability Matrices......Page 263
. Differential Equations and Characteristic Values......Page 264
. Effective Tests for Stability Matrices......Page 265
. A Simple Stochastic Process......Page 271
. Analytic Formulation of Discrete Markoff Processes......Page 273
. First Proof......Page 274
. Second Proof of Independence of Initial State......Page 276
. Some Properties of Positive Markoff Matrices......Page 277
. Second Proof of Limiting Behavior......Page 278
. General Markoff Matrices......Page 279
. A Continuous Stochastic Process......Page 280
. Generalized Probabilities-Unitary Transformations......Page 282
. Generalized Probabilities-Matrix Transformations......Page 283
. Limiting Behavior of Physical Systems......Page 289
. Expected Values......Page 290
. Expected Values of Squares......Page 291
. Some Simple Growth Processes......Page 294
. Notation......Page 295
. Proof of Theorem 1......Page 296
. Second Proof of the Simplicity of lambda(A)......Page 298
. Proof of the Minimum Property of lambd(A)......Page 299
. Steady-state Growth......Page 300
. Analogue of Perron Theorem......Page 301
. Mathematical Economics......Page 302
. Positivity of |I - A|......Page 306
. Linear Programming......Page 307
. The Theory of Games......Page 308
. A Markovian Decision Process......Page 309
. An Economic Model......Page 310
. Determinants......Page 321
. Homogeneous Systems......Page 322
. Rank......Page 326
. Signature......Page 327
Appendix B. The Quadratic Form of Selberg......Page 329
Appendix C. A Method of Hermite......Page 333
. A Device of Stieltjes......Page 335
. A Technique of E. Fischer......Page 336
. Representation as Moments......Page 337
. A Result of Herglotz......Page 338
Index......Page 341