Matrices: methods and applications

This volume provides a down-to-earth, easily understandable guide to techniques of matrix theory, which are widely used throughout engineering and the physical, life, and social sciences. Fully up-to-date, the book covers a wide range of topics, from basic matrix algebra to such advanced concepts as generalized inverses and Hadamard matrices, and applications to error-correcting codes, control theory, and linear programming. Results are illustrated with many examples drawn from diverse areas of application. Numerous exercises are included to clarify the material presented in the text, which is suitable for undergraduates and graduates alike. Researchers will also benefit from the accessible accounts of advanced matrix techniques. Barnett,S.Matrices_methods and applications(Oxford applied mathematics and computing science series;v.13)(OUP,1996)(ISBN O19859680)(600dpi)(466p) ......Page 4 Copyright ......Page 5 Preface v ......Page 6 Contents viii ......Page 9 Notation xv ......Page 14 1. How matrices arise 1 ......Page 16 Problems 7 ......Page 22 2.1 Definitions 11 ......Page 26 2.2.2 Multiplication by a scalar 12 ......Page 27 2.2.3 Multiplication of two matrices 15 ......Page 30 2.3.1 Definition and properties 22 ......Page 37 2.3.2 Symmetric and hermitian matrices 25 ......Page 40 2.4 Partitioning and submatrices 27 ......Page 42 2.5 Kronecker and Hadamard products 29 ......Page 44 2.6 Derivative of a matrix 32 ......Page 47 Problems 33 ......Page 48 3. Unique solution of linear equations 37 ......Page 52 3.1 Two equations and unknowns 38 ......Page 53 3.2 Gaussian elimination 39 ......Page 54 3.3 Triangular decomposition 44 ......Page 59 3.4 Ill-conditioning 49 ......Page 64 Problems 50 ......Page 65 4. Determinant and inverse 53 ......Page 68 4.1.1 3 x 3 case 54 ......Page 69 4.1.2 General properties 57 ......Page 72 4.1.3 Some applications 62 ......Page 77 4.2 Evaluation of determinants 64 ......Page 79 4.3.1 Definition and properties 67 ......Page 82 4.3.2 Partitioned form 71 ......Page 86 4.4 Calculation of inverse 72 ......Page 87 4.5 Cramer’s rule 78 ......Page 93 Problems 79 ......Page 94 5.1 Unique solution 86 ......Page 101 5.2 Definition of rank 87 ......Page 102 5.3.1 Elementary operations 88 ......Page 103 5.3.2 Calculation of rank 91 ......Page 106 5.3.3 Normal form 94 ......Page 109 5.4.1 Homogeneous equations 96 ......Page 111 5.4.2 Inhomogeneous equations 99 ......Page 114 5.4.3 Consistency theorem 103 ......Page 118 5.5 Method of least squares 104 ......Page 119 5.6 Use of Kronecker product 108 ......Page 123 5.7 Linear dependence of vectors 110 ......Page 125 5.8 Error-correcting codes 113 ......Page 128 Problems 120 ......Page 135 6.1 Definitions 127 ......Page 142 6.2 Some applications 130 ......Page 145 6.3.1 The characteristic equation 133 ......Page 148 6.3.2 Hermitian and symmetric matrices 136 ......Page 151 6.3.3 Matrix polynomials and the Cayley-Hamilton theorem 137 ......Page 152 6.3.4 Companion matrix 142 ......Page 157 6.3.5 Kronecker product expressions 146 ......Page 161 6.4.1 Definition 147 ......Page 162 6.4.2 Diagonalization 148 ......Page 163 6.4.3 Hermitian and symmetric matrices 150 ......Page 165 6.4.4 Transformation to companion form 152 ......Page 167 6.5 Solution of linear differential and difference equations 154 ......Page 169 6.6.1 Power method 156 ......Page 171 6.6.2 Other methods 158 ......Page 173 6.7.1 Gauss-Seidel and Jacobi methods 159 ......Page 174 6.7.2 Newton-Raphson type method 165 ......Page 180 Problems 167 ......Page 182 7. Quadratic and hermitian forms 175 ......Page 190 7.1 Definitions 176 ......Page 191 7.2 Lagrange’s reduction of quadratic forms 180 ......Page 195 7.3 Sylvester’s law of inertia 183 ......Page 198 7.4.1 Definitions 184 ......Page 199 7.4.2 Tests 185 ......Page 200 7.5.2 Optimization of functions 190 ......Page 205 7.5.3 Rayleigh quotient 192 ......Page 207 7.5.4 Liapunov stability 194 ......Page 209 Problems 195 ......Page 210 8. Canonical forms 198 ......Page 213 8.1 Jordan form 199 ......Page 214 8.2 Normal forms 206 ......Page 221 8.3 Schur form 209 ......Page 224 8.4 Hessenberg form 214 ......Page 229 8.5 Singular value and polar decompositions 218 ......Page 233 Problems 222 ......Page 237 9.1 Definition and properties 226 ......Page 241 9.2 Sylvester’s formula 233 ......Page 248 9.3 Linear differential and difference equations 239 ......Page 254 9.4 Matrix sign function 241 ......Page 256 Problems 244 ......Page 259 10.1.1 Definition 248 ......Page 263 10.1.2 Properties 251 ......Page 266 10.1.3 Computation 254 ......Page 269 10.2 Other inverses 260 ......Page 275 10.2.1 (i, j, k) inverses 261 ......Page 276 10.2.2 Drazin inverse 267 ......Page 282 10.3 Solution of linear equations 270 ......Page 285 10.4.1 Linear feedback control 273 ......Page 288 10.4.2 Singular systems 274 ......Page 289 10.4.4 Estimation of parameters 275 ......Page 290 Problems 276 ......Page 291 11.1 Companion matrices 280 ......Page 295 11.2 Resultant matrices 282 ......Page 297 11.2.2 Sylvester matrix 283 ......Page 298 11.2.3 Bezoutian matrix 284 ......Page 299 11.3.1 Computation via row operations 287 ......Page 302 11.3.2 Euclid’s algorithm 291 ......Page 306 11.3.3 Diophantine equations 294 ......Page 309 11.4.1 Relative to the imaginary axis 297 ......Page 312 11.4.2 Relative to the unit circle 301 ......Page 316 11.4.3 Bilinear transformation 304 ......Page 319 11.5.1 Liapunov equations 307 ......Page 322 11.5.2 Riccati equation 310 ......Page 325 11.5.3 Solution via eigenvectors 312 ......Page 327 11.5.4 Solution via matrix sign function 314 ......Page 329 Problems 317 ......Page 332 12. Polynomial and rational matrices 322 ......Page 337 12.1 Basic properties of polynomial matrices 323 ......Page 338 12.2 Elementary operations and Smith normal form 326 ......Page 341 12.3.1 Relative primeness 331 ......Page 346 12.3.2 Skew primeness 333 ......Page 348 12.4.1 Smith-McMillan form 335 ......Page 350 12.4.2 Transfer function matrices 337 ......Page 352 Problems 340 ......Page 355 13. Patterned matrices 342 ......Page 357 13.1 Banded matrices 343 ......Page 358 13.2 Circulant matrices 350 ......Page 365 13.3 Toeplitz and Hankel matrices 355 ......Page 370 13.4.1 Brownian matrices 367 ......Page 382 13.4.2 Centrosymmetric matrices 368 ......Page 383 13.4.3 Comrade matrix 369 ......Page 384 13.4.4 Loewner matrix 372 ......Page 387 13.4.5 Permutation matrices 373 ......Page 388 13.4.6 Sequence Hankel matrices 377 ......Page 392 Problems 378 ......Page 393 14.1.1 AX = XB 385 ......Page 400 14.1.2 Commuting matrices 387 ......Page 402 14.1.3 f(X) = 0 388 ......Page 403 14.1.4 Other equations 389 ......Page 404 14.2.1 Basic properties 390 ......Page 405 14.2.2 M-matrices 391 ......Page 406 14.2.3 Stochastic matrices 394 ......Page 409 14.2.4 Other forms 395 ......Page 410 14.3.1 Vector norms 396 ......Page 411 14.3.2 Matrix norms 397 ......Page 412 14.3.3 Conditioning 399 ......Page 414 14.4.1 Hadamard matrices 401 ......Page 416 14.4.2 Inequalities 405 ......Page 420 14.4.3 Interval matrices 406 ......Page 421 14.4.4 Unimodular integer matrices 409 ......Page 424 Exercises 412 ......Page 427 Problems 416 ......Page 431 References for chapters 421 ......Page 436 Some additional references on applications 423 ......Page 438 Answers to exercises 425 ......Page 440 Answers to problems 438 ......Page 453 Index 443 ......Page 458 cover......Page 1