Verfügbarkeit: Linear algebra in action

Linear algebra in action:

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Bibliographische Detailangaben
1. Verfasser:	Dym, Harry 1938- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Providence, R.I. American Math. Soc. 2007
Schriftenreihe:	Graduate studies in mathematics 78
Schlagworte:	Algèbre linéaire - Manuels d'enseignement supérieur Algebras, Linear Lineare Algebra
Online-Zugang:	Inhaltsverzeichnis Inhaltsverzeichnis Inhaltsverzeichnis
Beschreibung:	XVI, 541 S. graph. Darst. 27 cm
ISBN:	082183813X 9780821838136

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MARC


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Datensatz im Suchindex

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adam_text	LINEAR ALGEBRA IN ACTION HARRY DYM GRADUATE STUDIES IN MATHEMATICS VOLUME 78 AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND CONTENTS PREFACE XV CHAPTER 1. VECTOR SPACES 1 §1.1. PREVIEW 1 §1.2. THE ABSTRACT DEFINITION OF A VECTOR SPACE 2 §1.3. SOME DEFINITIONS 5 §1.4. MAPPINGS 11 §1.5. TRIANGULAR MATRICES 13 §1.6. BLOCK TRIANGULAR MATRICES 16 §1.7. SCHUR COMPLEMENTS 17 §1.8. OTHER MATRIX PRODUCTS 19 CHAPTER 2. GAUSSIAN ELIMINATION 21 §2.1. SOME PRELIMINARY OBSERVATIONS 22 §2.2. EXAMPLES 24 §2.3. UPPER ECHELON MATRICES 30 §2.4. THE CONSERVATION OF DIMENSION 36 §2.5. QUOTIENT SPACES 38 §2.6. CONSERVATION OF DIMENSION FOR MATRICES 38 §2.7. FROM U TO A 40 §2.8. SQUARE MATRICES 41 CHAPTER 3. ADDITIONAL APPLICATIONS OF GAUSSIAN ELIMINATION 45 §3.1. GAUSSIAN ELIMINATION REDUX 45 VI CONTENTS §3.2. PROPERTIES OF BA AND AC 48 §3.3. EXTRACTING A BASIS 50 §3.4. COMPUTING THE COEFFICIENTS IN A BASIS 51 §3.5. THE GAUSS-SEIDEL METHOD 52 §3.6. BLOCK GAUSSIAN ELIMINATION 55 §3.7. {0, 1, OO} 56 §3.8. REVIEW 57 CHAPTER 4. EIGENVALUES AND EIGENVECTORS 61 §4.1. CHANGE OF BASIS AND SIMILARITY 62 §4.2. INVARIANT SUBSPACES 64 §4.3. EXISTENCE OF EIGENVALUES 64 §4.4. EIGENVALUES FOR MATRICES 66 §4.5. DIRECT SUMS 69 §4.6. DIAGONALIZABLE MATRICES 71 §4.7. AN ALGORITHM FOR DIAGONALIZING MATRICES 73 §4.8. COMPUTING EIGENVALUES AT THIS POINT 74 §4.9. NOT ALL MATRICES ARE DIAGONALIZABLE 76 §4.10. THE JORDAN DECOMPOSITION THEOREM 78 §4.11. AN INSTRUCTIVE EXAMPLE 79 §4.12. THE BINOMIAL FORMULA 82 §4.13. MORE DIRECT SUM DECOMPOSITIONS 82 §4.14. VERIFICATION OF THEOREM 4.12 84 §4.15. BIBLIOGRAPHICAL NOTES 87 CHAPTER 5. DETERMINANTS 89 §5.1. FUNCTIONALS 89 §5.2. DETERMINANTS 90 §5.3. USEFUL RULES FOR CALCULATING DETERMINANTS 93 §5.4. EIGENVALUES 97 §5.5. EXPLOITING BLOCK STRUCTURE 99 §5.6. THE BINET-CAUCHY FORMULA 102 §5.7. MINORS 104 §5.8. USES OF DETERMINANTS 108 §5.9. COMPANION MATRICES 108 §5.10. CIRCULANTS AND VANDERMONDE MATRICES 109 CONTENTS . VII CHAPTER 6. CALCULATING JORDAN FORMS 111 §6.1. OVERVIEW 112 §6.2. STRUCTURE OF THE NULLSPACES J F BJ 112 §6.3. CHAINS AND CELLS 115 §6.4. COMPUTING J 116 §6.5. AN ALGORITHM FOR U 117 §6.6. AN EXAMPLE 120 §6.7. ANOTHER EXAMPLE 122 §6.8. JORDAN DECOMPOSITIONS FOR REAL MATRICES 126 §6.9. COMPANION AND GENERALIZED VANDERMONDE MATRICES 128 CHAPTER 7. NORMED LINEAR SPACES 133 §7.1. FOUR INEQUALITIES 133 §7.2. NORMED LINEAR SPACES 138 §7.3. EQUIVALENCE OF NORMS 140 §7.4. NORMS OF LINEAR TRANSFORMATIONS 142 §7.5. MULTIPLICATIVE NORMS 143 §7.6. EVALUATING SOME OPERATOR NORMS 145 §7.7. SMALL PERTURBATIONS 147 §7.8. ANOTHER ESTIMATE 149 §7.9. BOUNDED LINEAR FUNCTIONAL 150 §7.10. EXTENSIONS OF BOUNDED LINEAR FUNCTIONALS 152 §7.11. BANACH SPACES 155 CHAPTER 8. INNER PRODUCT SPACES AND ORTHOGONALITY 157 §8.1. INNER PRODUCT SPACES 157 §8.2. A CHARACTERIZATION OF INNER PRODUCT SPACES 160 §8.3. ORTHOGONALITY 161 §8.4. GRAM MATRICES 163 §8.5. ADJOINTS 163 §8.6. THE RIESZ REPRESENTATION THEOREM 166 §8.7. NORMAL, SELFADJOINT AND UNITARY TRANSFORMATIONS 168 §8.8. PROJECTIONS AND DIRECT SUM DECOMPOSITIONS 170 §8.9. ORTHOGONAL PROJECTIONS 172 §8.10. ORTHOGONAL EXPANSIONS 174 §8.11. THE GRAM-SCHMIDT METHOD 177 VIII CONTENTS §8.12. TOEPLITZ AND HANKEL MATRICES 178 §8.13. GAUSSIAN QUADRATURE 180 §8.14. BIBLIOGRAPHICAL NOTES 183 CHAPTER 9. SYMMETRIC, HERMITIAN AND NORMAL MATRICES 185 §9.1. HERMITIAN MATRICES ARE DIAGONALIZABLE 186 §9.2. COMMUTING HERMITIAN MATRICES 188 §9.3. REAL HERMITIAN MATRICES 190 §9.4. PROJECTIONS AND DIRECT SUMS IN F N 191 §9.5. PROJECTIONS AND RANK 195 §9.6. NORMAL MATRICES 195 §9.7. SCHUR S THEOREM 198 §9.8. QR FACTORIZATION 201 §9.9. AREAS, VOLUMES AND DETERMINANTS 202 §9.10. BIBLIOGRAPHICAL NOTES 206 CHAPTER 10. SINGULAR VALUES AND RELATED INEQUALITIES 207 §10.1. SINGULAR VALUE DECOMPOSITIONS 207 §10.2. COMPLEX SYMMETRIC MATRICES 212 §10.3. APPROXIMATE SOLUTIONS OF LINEAR EQUATIONS 213 §10.4. THE COURANT-FISCHER THEOREM 215 §10.5. INEQUALITIES FOR SINGULAR VALUES 218 §10.6. BIBLIOGRAPHICAL NOTES 225 CHAPTER 11. PSEUDOINVERSES 227 §11.1. PSEUDOINVERSES 227 §11.2. THE MOORE-PENROSE INVERSE 234 §11.3. BEST APPROXIMATION IN TERMS OF MOORE-PENROSE INVERSES 237 CHAPTER 12. TRIANGULAR FACTORIZATION AND POSITIVE DEFINITE MATRICES 239 §12.1. A DETOUR ON TRIANGULAR FACTORIZATION 240 §12.2. DEFINITE AND SEMIDEFMITE MATRICES 242 §12.3. CHARACTERIZATIONS OF POSITIVE DEFINITE MATRICES 244 §12.4. AN APPLICATION OF FACTORIZATION 247 §12.5. POSITIVE DEFINITE TOEPLITZ MATRICES 248 §12.6. DETOUR ON BLOCK TOEPLITZ MATRICES 254 §12.7. A MAXIMUM ENTROPY MATRIX COMPLETION PROBLEM 258 §12.8. SCHUR COMPLEMENTS FOR SEMIDEFINITE MATRICES 262 CONTENTS IX §12.9. SQUARE ROOTS 265 §12.10. POLAR FORMS 267 §12.11. MATRIX INEQUALITIES 268 §12.12. A MINIMAL NORM COMPLETION PROBLEM 271 §12.13. A DESCRIPTION OF ALL SOLUTIONS TO THE MINIMAL NORM COMPLETION PROBLEM 273 §12.14. BIBLIOGRAPHICAL NOTES 274 CHAPTER 13. DIFFERENCE EQUATIONS AND DIFFERENTIAL EQUATIONS 275 §13.1. SYSTEMS OF DIFFERENCE EQUATIONS 276 §13.2. THE EXPONENTIAL E TA 277 §13.3. SYSTEMS OF DIFFERENTIAL EQUATIONS 279 §13.4. UNIQUENESS 281 §13.5. ISOMETRIC AND ISOSPECTRAL FLOWS 282 §13.6. SECOND-ORDER DIFFERENTIAL SYSTEMS 283 §13.7. STABILITY 284 §13.8. NONHOMOGENEOUS DIFFERENTIAL SYSTEMS 285 §13.9. STRATEGY FOR EQUATIONS 285 §13.10. SECOND-ORDER DIFFERENCE EQUATIONS 286 §13.11. HIGHER ORDER DIFFERENCE EQUATIONS 289 §13.12. ORDINARY DIFFERENTIAL EQUATIONS 290 §13.13. WRONSKIANS 293 §13.14. VARIATION OF PARAMETERS 295 CHAPTER 14. VECTOR VALUED FUNCTIONS 297 §14.1. MEAN VALUE THEOREMS 298 §14.2. TAYLOR S FORMULA WITH REMAINDER 299 §14.3. APPLICATION OF TAYLOR S FORMULA WITH REMAINDER 300 §14.4. MEAN VALUE THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES 301 §14.5. MEAN VALUE THEOREMS FOR VECTOR VALUED FUNCTIONS OF SEVERAL VARIABLES 301 §14.6. NEWTON S METHOD 304 §14.7. A CONTRACTIVE FIXED POINT THEOREM 306 §14.8. A REFINED CONTRACTIVE FIXED POINT THEOREM 308 §14.9. SPECTRAL RADIUS 309 §14.10. THE BROUWER FIXED POINT THEOREM 313 §14.11. BIBLIOGRAPHICAL NOTES 316 X . CONTENTS CHAPTER 15. THE IMPLICIT FUNCTION THEOREM 317 §15.1. PRELIMINARY DISCUSSION 317 §15.2. THE MAIN THEOREM 319 §15.3. A GENERALIZATION OF THE IMPLICIT FUNCTION THEOREM 324 §15.4. CONTINUOUS DEPENDENCE OF SOLUTIONS 326 §15.5. THE INVERSE FUNCTION THEOREM 327 §15.6. ROOTS OF POLYNOMIALS 329 §15.7. AN INSTRUCTIVE EXAMPLE 329 §15.8. A MORE SOPHISTICATED APPROACH 331 §15.9. DYNAMICAL SYSTEMS 333 §15.10. LYAPUNOV FUNCTIONS 335 §15.11. BIBLIOGRAPHICAL NOTES 336 CHAPTER 16. EXTREMAL PROBLEMS 337 §16.1. CLASSICAL EXTREMAL PROBLEMS 337 §16.2. EXTREMAL PROBLEMS WITH CONSTRAINTS 341 §16.3. EXAMPLES 344 §16.4. KRYLOV SUBSPACES 349 §16.5. THE CONJUGATE GRADIENT METHOD 349 §16.6. DUAL EXTREMAL PROBLEMS 354 §16.7. BIBLIOGRAPHICAL NOTES 356 CHAPTER 17. MATRIX VALUED HOLOMORPHIC FUNCTIONS 357 §17.1. DIFFERENTIATION 357 §17.2. CONTOUR INTEGRATION 361 §17.3. EVALUATING INTEGRALS BY CONTOUR INTEGRATION 365 §17.4. A SHORT DETOUR ON FOURIER ANALYSIS 370 §17.5. CONTOUR INTEGRALS OF MATRIX VALUED FUNCTIONS 372 §17.6. CONTINUOUS DEPENDENCE OF THE EIGENVALUES 375 §17.7. MORE ON SMALL PERTURBATIONS 377 §17.8. SPECTRAL RADIUS REDUX 378 §17.9. FRACTIONAL POWERS 381 CHAPTER 18. MATRIX EQUATIONS 383 §18.1. THE EQUATION X - AXB = C 38 3 §18.2. THE SYLVESTER EQUATION AX - XB = C 385 §18.3. SPECIAL CLASSES OF SOLUTIONS 388 CONTENTS . XI §18.4. §18.5. §18.6. §18.7. CHAPTER §19.1. §19.2. §19.3. §19.4. §19.5. §19.6. §19.7. CHAPTER §20.1. §20.2. §20.3. §20.4. §20.5. §20.6. §20.7. §20.8. §20.9. §20.10 §20.11 CHAPTER §21.1. §21.2. §21.3. §21.4. §21.5. §21.6. §21.7. §21.8. 521.9. RICCATI EQUATIONS TWO LEMMAS AN LQR PROBLEM BIBLIOGRAPHICAL NOTES 19. REALIZATION THEORY MINIMAL REALIZATIONS STABILIZABLE AND DETECTABLE REALIZATIONS REPRODUCING KERNEL HILBERT SPACES DE BRANGES SPACES R A INVARIANCE FACTORIZATION OF (A) BIBLIOGRAPHICAL NOTES 20. .EIGENVALUE LOCATION PROBLEMS INTERLACING SYLVESTER S LAW OF INERTIA CONGRUENCE COUNTING POSITIVE AND NEGATIVE EIGENVALUES EXPLOITING CONTINUITY GERSGORIN DISKS THE SPECTRAL MAPPING PRINCIPLE AX = XB INERTIA THEOREMS . AN EIGENVALUE ASSIGNMENT PROBLEM . BIBLIOGRAPHICAL NOTES 21. ZERO LOCATION PROBLEMS BEZOUTIANS A DERIVATION OF THE FORMULA FOR HF BASED ON REALIZATION THE BARNETT IDENTITY THE MAIN THEOREM ON BEZOUTIANS RESULTANTS OTHER DIRECTIONS BEZOUTIANS FOR REAL POLYNOMIALS STABLE POLYNOMIALS KHARITONOV S THEOREM 390 396 398 400 401 408 415 416 418 420 421 425 427 427 430 431 433 437 438 439 440 441 443 446 447 447 452 453 455 457 461 463 464 466 XII CONTENTS §21.10. BIBLIOGRAPHICAL NOTES 467 CHAPTER 22. CONVEXITY 469 §22.1. PRELIMINARIES 469 §22.2. CONVEX FUNCTIONS 471 §22.3. CONVEX SETS IN R N 473 §22.4. SEPARATION THEOREMS IN R N 475 §22.5. HYPERPLANES 477 §22.6. SUPPORT HYPERPLANES 479 §22.7. CONVEX HULLS 480 §22.8. EXTREME POINTS 482 §22.9. BROUWER S THEOREM FOR COMPACT CONVEX SETS 485 §22.10. THE MINKOWSKI FUNCTIONAL 485 §22.11. THE GAUSS-LUCAS THEOREM 488 §22.12. THE NUMERICAL RANGE 489 §22.13. EIGENVALUES VERSUS NUMERICAL RANGE 491 §22.14. THE HEINZ INEQUALITY 492 §22.15. BIBLIOGRAPHICAL NOTES 494 CHAPTER 23. MATRICES WITH NONNEGATIVE ENTRIES 495 §23.1. PERRON-FROBENIUS THEORY 496 §23.2. STOCHASTIC MATRICES 503 §23.3. DOUBLY STOCHASTIC MATRICES 504 §23.4. AN INEQUALITY OF KY FAN 507 §23.5. THE SCHUR-HORN CONVEXITY THEOREM 509 §23.6. BIBLIOGRAPHICAL NOTES 513 APPENDIX A. SOME FACTS FROM ANALYSIS 515 §A.L. CONVERGENCE OF SEQUENCES OF POINTS 515 §A.2. CONVERGENCE OF SEQUENCES OF FUNCTIONS 516 §A.3. CONVERGENCE OF SUMS 516 §A.4. SUPS AND INFS 517 §A.5. TOPOLOGY 518 §A.6. COMPACT SETS 518 §A.7. NORMED LINEAR SPACES 518 APPENDIX B. MORE COMPLEX VARIABLES 521 §B.L. POWER SERIES 521 CONTENTS - XIII §B.2. ISOLATED ZEROS 523 §B.3. THE MAXIMUM MODULUS PRINCIPLE 525 §B.4. IN (1 - A) WHEN \|A\| 1 525 §B.5. ROUCHE S THEOREM 526 §B.6. LIOUVILLE S THEOREM 528 §B.7. LAURENT EXPANSIONS 528 §B.8. PARTIAL FRACTION EXPANSIONS 529 BIBLIOGRAPHY 531 NOTATION INDEX 535 SUBJECT INDEX 537
adam_txt	LINEAR ALGEBRA IN ACTION HARRY DYM GRADUATE STUDIES IN MATHEMATICS VOLUME 78 AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND CONTENTS PREFACE XV CHAPTER 1. VECTOR SPACES 1 §1.1. PREVIEW 1 §1.2. THE ABSTRACT DEFINITION OF A VECTOR SPACE 2 §1.3. SOME DEFINITIONS 5 §1.4. MAPPINGS 11 §1.5. TRIANGULAR MATRICES 13 §1.6. BLOCK TRIANGULAR MATRICES 16 §1.7. SCHUR COMPLEMENTS 17 §1.8. OTHER MATRIX PRODUCTS 19 CHAPTER 2. GAUSSIAN ELIMINATION 21 §2.1. SOME PRELIMINARY OBSERVATIONS 22 §2.2. EXAMPLES 24 §2.3. UPPER ECHELON MATRICES 30 §2.4. THE CONSERVATION OF DIMENSION 36 §2.5. QUOTIENT SPACES 38 §2.6. CONSERVATION OF DIMENSION FOR MATRICES 38 §2.7. FROM U TO A 40 §2.8. SQUARE MATRICES 41 CHAPTER 3. ADDITIONAL APPLICATIONS OF GAUSSIAN ELIMINATION 45 §3.1. GAUSSIAN ELIMINATION REDUX 45 VI CONTENTS §3.2. PROPERTIES OF BA AND AC 48 §3.3. EXTRACTING A BASIS 50 §3.4. COMPUTING THE COEFFICIENTS IN A BASIS 51 §3.5. THE GAUSS-SEIDEL METHOD 52 §3.6. BLOCK GAUSSIAN ELIMINATION 55 §3.7. {0, 1, OO} 56 §3.8. REVIEW 57 CHAPTER 4. EIGENVALUES AND EIGENVECTORS 61 §4.1. CHANGE OF BASIS AND SIMILARITY 62 §4.2. INVARIANT SUBSPACES 64 §4.3. EXISTENCE OF EIGENVALUES 64 §4.4. EIGENVALUES FOR MATRICES 66 §4.5. DIRECT SUMS 69 §4.6. DIAGONALIZABLE MATRICES 71 §4.7. AN ALGORITHM FOR DIAGONALIZING MATRICES 73 §4.8. COMPUTING EIGENVALUES AT THIS POINT 74 §4.9. NOT ALL MATRICES ARE DIAGONALIZABLE 76 §4.10. THE JORDAN DECOMPOSITION THEOREM 78 §4.11. AN INSTRUCTIVE EXAMPLE 79 §4.12. THE BINOMIAL FORMULA 82 §4.13. MORE DIRECT SUM DECOMPOSITIONS 82 §4.14. VERIFICATION OF THEOREM 4.12 84 §4.15. BIBLIOGRAPHICAL NOTES 87 CHAPTER 5. DETERMINANTS 89 §5.1. FUNCTIONALS 89 §5.2. DETERMINANTS 90 §5.3. USEFUL RULES FOR CALCULATING DETERMINANTS 93 §5.4. EIGENVALUES 97 §5.5. EXPLOITING BLOCK STRUCTURE 99 §5.6. THE BINET-CAUCHY FORMULA 102 §5.7. MINORS 104 §5.8. USES OF DETERMINANTS 108 §5.9. COMPANION MATRICES 108 §5.10. CIRCULANTS AND VANDERMONDE MATRICES 109 CONTENTS . VII CHAPTER 6. CALCULATING JORDAN FORMS 111 §6.1. OVERVIEW 112 §6.2. STRUCTURE OF THE NULLSPACES J\F BJ 112 §6.3. CHAINS AND CELLS 115 §6.4. COMPUTING J 116 §6.5. AN ALGORITHM FOR U 117 §6.6. AN EXAMPLE 120 §6.7. ANOTHER EXAMPLE 122 §6.8. JORDAN DECOMPOSITIONS FOR REAL MATRICES 126 §6.9. COMPANION AND GENERALIZED VANDERMONDE MATRICES 128 CHAPTER 7. NORMED LINEAR SPACES 133 §7.1. FOUR INEQUALITIES 133 §7.2. NORMED LINEAR SPACES 138 §7.3. EQUIVALENCE OF NORMS 140 §7.4. NORMS OF LINEAR TRANSFORMATIONS 142 §7.5. MULTIPLICATIVE NORMS 143 §7.6. EVALUATING SOME OPERATOR NORMS 145 §7.7. SMALL PERTURBATIONS 147 §7.8. ANOTHER ESTIMATE 149 §7.9. BOUNDED LINEAR FUNCTIONAL 150 §7.10. EXTENSIONS OF BOUNDED LINEAR FUNCTIONALS 152 §7.11. BANACH SPACES 155 CHAPTER 8. INNER PRODUCT SPACES AND ORTHOGONALITY 157 §8.1. INNER PRODUCT SPACES 157 §8.2. A CHARACTERIZATION OF INNER PRODUCT SPACES 160 §8.3. ORTHOGONALITY 161 §8.4. GRAM MATRICES 163 §8.5. ADJOINTS 163 §8.6. THE RIESZ REPRESENTATION THEOREM 166 §8.7. NORMAL, SELFADJOINT AND UNITARY TRANSFORMATIONS 168 §8.8. PROJECTIONS AND DIRECT SUM DECOMPOSITIONS 170 §8.9. ORTHOGONAL PROJECTIONS 172 §8.10. ORTHOGONAL EXPANSIONS 174 §8.11. THE GRAM-SCHMIDT METHOD 177 VIII CONTENTS §8.12. TOEPLITZ AND HANKEL MATRICES 178 §8.13. GAUSSIAN QUADRATURE 180 §8.14. BIBLIOGRAPHICAL NOTES 183 CHAPTER 9. SYMMETRIC, HERMITIAN AND NORMAL MATRICES 185 §9.1. HERMITIAN MATRICES ARE DIAGONALIZABLE 186 §9.2. COMMUTING HERMITIAN MATRICES 188 §9.3. REAL HERMITIAN MATRICES 190 §9.4. PROJECTIONS AND DIRECT SUMS IN F N 191 §9.5. PROJECTIONS AND RANK 195 §9.6. NORMAL MATRICES 195 §9.7. SCHUR'S THEOREM 198 §9.8. QR FACTORIZATION 201 §9.9. AREAS, VOLUMES AND DETERMINANTS 202 §9.10. BIBLIOGRAPHICAL NOTES 206 CHAPTER 10. SINGULAR VALUES AND RELATED INEQUALITIES 207 §10.1. SINGULAR VALUE DECOMPOSITIONS 207 §10.2. COMPLEX SYMMETRIC MATRICES 212 §10.3. APPROXIMATE SOLUTIONS OF LINEAR EQUATIONS 213 §10.4. THE COURANT-FISCHER THEOREM 215 §10.5. INEQUALITIES FOR SINGULAR VALUES 218 §10.6. BIBLIOGRAPHICAL NOTES 225 CHAPTER 11. PSEUDOINVERSES 227 §11.1. PSEUDOINVERSES 227 §11.2. THE MOORE-PENROSE INVERSE 234 §11.3. BEST APPROXIMATION IN TERMS OF MOORE-PENROSE INVERSES 237 CHAPTER 12. TRIANGULAR FACTORIZATION AND POSITIVE DEFINITE MATRICES 239 §12.1. A DETOUR ON TRIANGULAR FACTORIZATION 240 §12.2. DEFINITE AND SEMIDEFMITE MATRICES 242 §12.3. CHARACTERIZATIONS OF POSITIVE DEFINITE MATRICES 244 §12.4. AN APPLICATION OF FACTORIZATION 247 §12.5. POSITIVE DEFINITE TOEPLITZ MATRICES 248 §12.6. DETOUR ON BLOCK TOEPLITZ MATRICES 254 §12.7. A MAXIMUM ENTROPY MATRIX COMPLETION PROBLEM 258 §12.8. SCHUR COMPLEMENTS FOR SEMIDEFINITE MATRICES 262 CONTENTS IX §12.9. SQUARE ROOTS 265 §12.10. POLAR FORMS 267 §12.11. MATRIX INEQUALITIES 268 §12.12. A MINIMAL NORM COMPLETION PROBLEM 271 §12.13. A DESCRIPTION OF ALL SOLUTIONS TO THE MINIMAL NORM COMPLETION PROBLEM 273 §12.14. BIBLIOGRAPHICAL NOTES 274 CHAPTER 13. DIFFERENCE EQUATIONS AND DIFFERENTIAL EQUATIONS 275 §13.1. SYSTEMS OF DIFFERENCE EQUATIONS 276 §13.2. THE EXPONENTIAL E TA 277 §13.3. SYSTEMS OF DIFFERENTIAL EQUATIONS 279 §13.4. UNIQUENESS 281 §13.5. ISOMETRIC AND ISOSPECTRAL FLOWS 282 §13.6. SECOND-ORDER DIFFERENTIAL SYSTEMS 283 §13.7. STABILITY 284 §13.8. NONHOMOGENEOUS DIFFERENTIAL SYSTEMS 285 §13.9. STRATEGY FOR EQUATIONS 285 §13.10. SECOND-ORDER DIFFERENCE EQUATIONS 286 §13.11. HIGHER ORDER DIFFERENCE EQUATIONS 289 §13.12. ORDINARY DIFFERENTIAL EQUATIONS 290 §13.13. WRONSKIANS 293 §13.14. VARIATION OF PARAMETERS 295 CHAPTER 14. VECTOR VALUED FUNCTIONS 297 §14.1. MEAN VALUE THEOREMS 298 §14.2. TAYLOR'S FORMULA WITH REMAINDER 299 §14.3. APPLICATION OF TAYLOR'S FORMULA WITH REMAINDER 300 §14.4. MEAN VALUE THEOREM FOR FUNCTIONS OF SEVERAL VARIABLES 301 §14.5. MEAN VALUE THEOREMS FOR VECTOR VALUED FUNCTIONS OF SEVERAL VARIABLES 301 §14.6. NEWTON'S METHOD 304 §14.7. A CONTRACTIVE FIXED POINT THEOREM 306 §14.8. A REFINED CONTRACTIVE FIXED POINT THEOREM 308 §14.9. SPECTRAL RADIUS 309 §14.10. THE BROUWER FIXED POINT THEOREM 313 §14.11. BIBLIOGRAPHICAL NOTES 316 X . CONTENTS CHAPTER 15. THE IMPLICIT FUNCTION THEOREM 317 §15.1. PRELIMINARY DISCUSSION 317 §15.2. THE MAIN THEOREM 319 §15.3. A GENERALIZATION OF THE IMPLICIT FUNCTION THEOREM 324 §15.4. CONTINUOUS DEPENDENCE OF SOLUTIONS 326 §15.5. THE INVERSE FUNCTION THEOREM 327 §15.6. ROOTS OF POLYNOMIALS 329 §15.7. AN INSTRUCTIVE EXAMPLE 329 §15.8. A MORE SOPHISTICATED APPROACH 331 §15.9. DYNAMICAL SYSTEMS 333 §15.10. LYAPUNOV FUNCTIONS 335 §15.11. BIBLIOGRAPHICAL NOTES 336 CHAPTER 16. EXTREMAL PROBLEMS 337 §16.1. CLASSICAL EXTREMAL PROBLEMS 337 §16.2. EXTREMAL PROBLEMS WITH CONSTRAINTS 341 §16.3. EXAMPLES 344 §16.4. KRYLOV SUBSPACES 349 §16.5. THE CONJUGATE GRADIENT METHOD 349 §16.6. DUAL EXTREMAL PROBLEMS 354 §16.7. BIBLIOGRAPHICAL NOTES 356 CHAPTER 17. MATRIX VALUED HOLOMORPHIC FUNCTIONS 357 §17.1. DIFFERENTIATION 357 §17.2. CONTOUR INTEGRATION 361 §17.3. EVALUATING INTEGRALS BY CONTOUR INTEGRATION 365 §17.4. A SHORT DETOUR ON FOURIER ANALYSIS 370 §17.5. CONTOUR INTEGRALS OF MATRIX VALUED FUNCTIONS 372 §17.6. CONTINUOUS DEPENDENCE OF THE EIGENVALUES 375 §17.7. MORE ON SMALL PERTURBATIONS 377 §17.8. SPECTRAL RADIUS REDUX 378 §17.9. FRACTIONAL POWERS 381 CHAPTER 18. MATRIX EQUATIONS 383 §18.1. THE EQUATION X - AXB = C 38 3 §18.2. THE SYLVESTER EQUATION AX - XB = C 385 §18.3. SPECIAL CLASSES OF SOLUTIONS 388 CONTENTS . XI §18.4. §18.5. §18.6. §18.7. CHAPTER §19.1. §19.2. §19.3. §19.4. §19.5. §19.6. §19.7. CHAPTER §20.1. §20.2. §20.3. §20.4. §20.5. §20.6. §20.7. §20.8. §20.9. §20.10 §20.11 CHAPTER §21.1. §21.2. §21.3. §21.4. §21.5. §21.6. §21.7. §21.8. 521.9. RICCATI EQUATIONS TWO LEMMAS AN LQR PROBLEM BIBLIOGRAPHICAL NOTES 19. REALIZATION THEORY MINIMAL REALIZATIONS STABILIZABLE AND DETECTABLE REALIZATIONS REPRODUCING KERNEL HILBERT SPACES DE BRANGES SPACES R A INVARIANCE FACTORIZATION OF (A) BIBLIOGRAPHICAL NOTES 20. .EIGENVALUE LOCATION PROBLEMS INTERLACING SYLVESTER'S LAW OF INERTIA CONGRUENCE COUNTING POSITIVE AND NEGATIVE EIGENVALUES EXPLOITING CONTINUITY GERSGORIN DISKS THE SPECTRAL MAPPING PRINCIPLE AX = XB INERTIA THEOREMS . AN EIGENVALUE ASSIGNMENT PROBLEM . BIBLIOGRAPHICAL NOTES 21. ZERO LOCATION PROBLEMS BEZOUTIANS A DERIVATION OF THE FORMULA FOR HF BASED ON REALIZATION THE BARNETT IDENTITY THE MAIN THEOREM ON BEZOUTIANS RESULTANTS OTHER DIRECTIONS BEZOUTIANS FOR REAL POLYNOMIALS STABLE POLYNOMIALS KHARITONOV'S THEOREM 390 396 398 400 401 408 415 416 418 420 421 425 427 427 430 431 433 437 438 439 440 441 443 446 447 447 452 453 455 457 461 463 464 466 XII CONTENTS §21.10. BIBLIOGRAPHICAL NOTES 467 CHAPTER 22. CONVEXITY 469 §22.1. PRELIMINARIES 469 §22.2. CONVEX FUNCTIONS 471 §22.3. CONVEX SETS IN R N 473 §22.4. SEPARATION THEOREMS IN R N 475 §22.5. HYPERPLANES 477 §22.6. SUPPORT HYPERPLANES 479 §22.7. CONVEX HULLS 480 §22.8. EXTREME POINTS 482 §22.9. BROUWER'S THEOREM FOR COMPACT CONVEX SETS 485 §22.10. THE MINKOWSKI FUNCTIONAL 485 §22.11. THE GAUSS-LUCAS THEOREM 488 §22.12. THE NUMERICAL RANGE 489 §22.13. EIGENVALUES VERSUS NUMERICAL RANGE 491 §22.14. THE HEINZ INEQUALITY 492 §22.15. BIBLIOGRAPHICAL NOTES 494 CHAPTER 23. MATRICES WITH NONNEGATIVE ENTRIES 495 §23.1. PERRON-FROBENIUS THEORY 496 §23.2. STOCHASTIC MATRICES 503 §23.3. DOUBLY STOCHASTIC MATRICES 504 §23.4. AN INEQUALITY OF KY FAN 507 §23.5. THE SCHUR-HORN CONVEXITY THEOREM 509 §23.6. BIBLIOGRAPHICAL NOTES 513 APPENDIX A. SOME FACTS FROM ANALYSIS 515 §A.L. CONVERGENCE OF SEQUENCES OF POINTS 515 §A.2. CONVERGENCE OF SEQUENCES OF FUNCTIONS 516 §A.3. CONVERGENCE OF SUMS 516 §A.4. SUPS AND INFS 517 §A.5. TOPOLOGY 518 §A.6. COMPACT SETS 518 §A.7. NORMED LINEAR SPACES 518 APPENDIX B. MORE COMPLEX VARIABLES 521 §B.L. POWER SERIES 521 CONTENTS - XIII §B.2. ISOLATED ZEROS 523 §B.3. THE MAXIMUM MODULUS PRINCIPLE 525 §B.4. IN (1 - A) WHEN \|A\| 1 525 §B.5. ROUCHE'S THEOREM 526 §B.6. LIOUVILLE'S THEOREM 528 §B.7. LAURENT EXPANSIONS 528 §B.8. PARTIAL FRACTION EXPANSIONS 529 BIBLIOGRAPHY 531 NOTATION INDEX 535 SUBJECT INDEX 537
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title	Linear algebra in action
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topic	Algèbre linéaire - Manuels d'enseignement supérieur Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd
topic_facet	Algèbre linéaire - Manuels d'enseignement supérieur Algebras, Linear Lineare Algebra
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