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