Matrix Mathematics: Theory, Facts, and Formulas: Second Edition

✍ Scribed by Dennis S. Bernstein

• Publisher: Princeton University Press
• Year: 2009
• Tongue: English
• Leaves: 1103
• Edition: 2
• Category: Library

✦ Synopsis

When first published in 2005, Matrix Mathematics quickly became the essential reference book for users of matrices in all branches of engineering, science, and applied mathematics. In this fully updated and expanded edition, the author brings together the latest results on matrix theory to make this the most complete, current, and easy-to-use book on matrices. Each chapter describes relevant background theory followed by specialized results. Hundreds of identities, inequalities, and matrix facts are stated clearly and rigorously with cross references, citations to the literature, and illuminating remarks. Beginning with preliminaries on sets, functions, and relations,Matrix Mathematics covers all of the major topics in matrix theory, including matrix transformations; polynomial matrices; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; vector and matrix norms; the matrix exponential and stability theory; and linear systems and control theory. Also included are a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. This significantly expanded edition of Matrix Mathematics features a wealth of new material on graphs, scalar identities and inequalities, alternative partial orderings, matrix pencils, finite groups, zeros of multivariable transfer functions, roots of polynomials, convex functions, and matrix norms. Covers hundreds of important and useful results on matrix theory, many never before available in any book Provides a list of symbols and a summary of conventions for easy use Includes an extensive collection of scalar identities and inequalities Features a detailed bibliography and author index with page references Includes an exhaustive subject index with cross-referencing

✦ Table of Contents

MATRIX MATHEMATICS: THEORY, FACTS, AND FORMULAS, 2ND ED.......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface to the Second Edition......Page 16
Preface to the First Edition......Page 18
Special Symbols......Page 22
Conventions, Notation, and Terminology......Page 34
1.1 Logic and Sets......Page 44
1.2 Functions......Page 46
1.3 Relations......Page 48
1.4 Graphs......Page 51
1.5 Facts on Logic, Sets, Functions, and Relations......Page 53
1.6 Facts on Graphs......Page 56
1.7 Facts on Binomial Identities and Sums......Page 57
1.8 Facts on Convex Functions......Page 64
1.9 Facts on Scalar Identities and Inequalities in One Variable......Page 65
1.10 Facts on Scalar Identities and Inequalities in Two Variables......Page 73
1.11 Facts on Scalar Identities and Inequalities in Three Variables......Page 82
1.12 Facts on Scalar Identities and Inequalities in Four Variables......Page 89
1.14 Facts on Scalar Identities and Inequalities in Eight Variables......Page 90
1.15 Facts on Scalar Identities and Inequalities in n Variables......Page 91
1.16 Facts on Scalar Identities and Inequalities in 2n Variables......Page 103
1.17 Facts on Scalar Identities and Inequalities in 3n Variables......Page 110
1.18 Facts on Scalar Identities and Inequalities in Complex Variables......Page 111
1.19 Facts on Trigonometric and Hyperbolic Identities......Page 117
1.20 Notes......Page 119
2.1 Matrix Algebra......Page 120
2.2 Transpose and Inner Product......Page 127
2.3 Convex Sets, Cones, and Subspaces......Page 132
2.4 Range and Null Space......Page 136
2.5 Rank and Defect......Page 138
2.6 Invertibility......Page 141
2.7 The Determinant......Page 145
2.8 Partitioned Matrices......Page 149
2.9 Facts on Polars, Cones, Dual Cones, Convex Hulls, and Subspaces......Page 153
2.10 Facts on Range, Null Space, Rank, and Defect......Page 158
2.11 Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices......Page 163
2.12 Facts on the Inner Product, Outer Product, Trace, and Matrix Powers......Page 169
2.13 Facts on the Determinant......Page 171
2.14 Facts on the Determinant of Partitioned Matrices......Page 175
2.15 Facts on Left and Right Inverses......Page 183
2.16 Facts on the Adjugate and Inverses......Page 184
2.17 Facts on the Inverse of Partitioned Matrices......Page 189
2.18 Facts on Commutators......Page 192
2.19 Facts on Complex Matrices......Page 194
2.20 Facts on Geometry......Page 197
2.21 Facts on Majorization......Page 205
2.22 Notes......Page 207
3.1 Matrix Classes......Page 208
3.2 Matrices Based on Graphs......Page 213
3.3 Lie Algebras and Groups......Page 214
3.4 Matrix Transformations......Page 216
3.5 Projectors, Idempotent Matrices, and Subspaces......Page 218
3.6 Facts on Group-Invertible and Range-Hermitian Matrices......Page 220
3.7 Facts on Normal, Hermitian, and Skew-Hermitian Matrices......Page 221
3.8 Facts on Commutators......Page 227
3.9 Facts on Linear Interpolation......Page 228
3.10 Facts on the Cross Product......Page 229
3.11 Facts on Unitary and Shifted-Unitary Matrices......Page 232
3.12 Facts on Idempotent Matrices......Page 241
3.13 Facts on Projectors......Page 249
3.14 Facts on Reflectors......Page 254
3.16 Facts on Tripotent Matrices......Page 255
3.17 Facts on Nilpotent Matrices......Page 256
3.18 Facts on Hankel and Toeplitz Matrices......Page 258
3.19 Facts on Hamiltonian and Symplectic Matrices......Page 259
3.20 Facts on Miscellaneous Types of Matrices......Page 260
3.21 Facts on Groups......Page 264
3.22 Facts on Quaternions......Page 268
3.23 Notes......Page 272
4.1 Polynomials......Page 274
4.2 Polynomial Matrices......Page 277
4.3 The Smith Decomposition and Similarity Invariants......Page 279
4.4 Eigenvalues......Page 282
4.5 Eigenvectors......Page 288
4.6 The Minimal Polynomial......Page 290
4.7 Rational Transfer Functions and the Smith-McMillan Decomposition......Page 292
4.8 Facts on Polynomials and Rational Functions......Page 296
4.9 Facts on the Characteristic and Minimal Polynomials......Page 303
4.10 Facts on the Spectrum......Page 308
4.11 Facts on Graphs and Nonnegative Matrices......Page 315
4.12 Notes......Page 324
5.2 Multicompanion Form......Page 326
5.3 Hypercompanion Form and Jordan Form......Page 330
5.4 Schur Decomposition......Page 335
5.5 Eigenstructure Properties......Page 338
5.6 Singular Value Decomposition......Page 344
5.7 Pencils and the Kronecker Canonical Form......Page 347
5.8 Facts on the Inertia......Page 350
5.9 Facts on Matrix Transformations for One Matrix......Page 354
5.10 Facts on Matrix Transformations for Two or More Matrices......Page 359
5.11 Facts on Eigenvalues and Singular Values for One Matrix......Page 364
5.12 Facts on Eigenvalues and Singular Values for Two or More Matrices......Page 376
5.14 Facts on Matrix Eigenstructure......Page 381
5.15 Facts on Matrix Factorizations......Page 388
5.16 Facts on Companion, Vandermonde, and Circulant Matrices......Page 395
5.17 Facts on Simultaneous Transformations......Page 401
5.18 Facts on the Polar Decomposition......Page 402
5.19 Facts on Additive Decompositions......Page 403
5.20 Notes......Page 404
6.1 Moore-Penrose Generalized Inverse......Page 406
6.2 Drazin Generalized Inverse......Page 410
6.3 Facts on the Moore-Penrose Generalized Inverse for One Matrix......Page 412
6.4 Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices......Page 420
6.5 Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices......Page 428
6.6 Facts on the Drazin and Group Generalized Inverses......Page 436
6.7 Notes......Page 441
7.1 Kronecker Product......Page 442
7.2 Kronecker Sum and Linear Matrix Equations......Page 445
7.3 Schur Product......Page 447
7.4 Facts on the Kronecker Product......Page 448
7.5 Facts on the Kronecker Sum......Page 452
7.6 Facts on the Schur Product......Page 456
7.7 Notes......Page 459
8.1 Positive-Semidefinite and Positive-Definite Orderings......Page 460
8.2 Submatrices......Page 462
8.3 Simultaneous Diagonalization......Page 465
8.4 Eigenvalue Inequalities......Page 467
8.5 Exponential, Square Root, and Logarithm of Hermitian Matrices......Page 473
8.6 Matrix Inequalities......Page 474
8.7 Facts on Range and Rank......Page 486
8.8 Facts on Structured Positive-Semidefinite Matrices......Page 487
8.9 Facts on Identities and Inequalities for One Matrix......Page 493
8.10 Facts on Identities and Inequalities for Two or More Matrices......Page 499
8.11 Facts on Identities and Inequalities for Partitioned Matrices......Page 510
8.12 Facts on the Trace......Page 518
8.13 Facts on the Determinant......Page 528
8.14 Facts on Convex Sets and Convex Functions......Page 537
8.15 Facts on Quadratic Forms......Page 543
8.16 Facts on Simultaneous Diagonalization......Page 550
8.17 Facts on Eigenvalues and Singular Values for One Matrix......Page 551
8.18 Facts on Eigenvalues and Singular Values for Two or More Matrices......Page 555
8.19 Facts on Alternative Partial Orderings......Page 565
8.20 Facts on Generalized Inverses......Page 568
8.21 Facts on the Kronecker and Schur Products......Page 574
8.22 Notes......Page 584
9.1 Vector Norms......Page 586
9.2 Matrix Norms......Page 589
9.3 Compatible Norms......Page 592
9.4 Induced Norms......Page 596
9.5 Induced Lower Bound......Page 601
9.6 Singular Value Inequalities......Page 603
9.7 Facts on Vector Norms......Page 606
9.8 Facts on Matrix Norms for One Matrix......Page 614
9.9 Facts on Matrix Norms for Two or More Matrices......Page 623
9.10 Facts on Matrix Norms for Partitioned Matrices......Page 636
9.11 Facts on Matrix Norms and Eigenvalues Involving One Matrix......Page 639
9.12 Facts on Matrix Norms and Eigenvalues Involving Two or More Matrices......Page 642
9.13 Facts on Matrix Norms and Singular Values for One Matrix......Page 645
9.14 Facts on Matrix Norms and Singular Values for Two or More Matrices......Page 650
9.15 Facts on Least Squares......Page 661
9.16 Notes......Page 662
10.1 Open Sets and Closed Sets......Page 664
10.2 Limits......Page 665
10.3 Continuity......Page 666
10.4 Derivatives......Page 668
10.5 Functions of a Matrix......Page 671
10.6 Matrix Square Root and Matrix Sign Functions......Page 672
10.7 Matrix Derivatives......Page 673
10.8 Facts Involving One Set......Page 675
10.9 Facts Involving Two or More Sets......Page 677
10.10 Facts on Matrix Functions......Page 680
10.11 Facts on Functions and Derivatives......Page 681
10.12 Notes......Page 685
11.1 Definition of the Matrix Exponential......Page 686
11.2 Structure of the Matrix Exponential......Page 689
11.3 Explicit Expressions......Page 694
11.4 Matrix Logarithms......Page 697
11.5 The Logarithm Function......Page 699
11.6 Lie Groups......Page 701
11.7 Lyapunov Stability Theory......Page 703
11.8 Linear Stability Theory......Page 705
11.9 The Lyapunov Equation......Page 709
11.10 Discrete-Time Stability Theory......Page 712
11.11 Facts on Matrix Exponential Formulas......Page 714
11.13 Facts on the Matrix Exponential for One Matrix......Page 720
11.14 Facts on the Matrix Exponential for Two or More Matrices......Page 724
11.15 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for One Matrix......Page 732
11.16 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for Two or More Matrices......Page 735
11.17 Facts on Stable Polynomials......Page 738
11.18 Facts on Stable Matrices......Page 741
11.19 Facts on Almost Nonnegative Matrices......Page 749
11.20 Facts on Discrete-Time-Stable Polynomials......Page 751
11.21 Facts on Discrete-Time-Stable Matrices......Page 755
11.22 Facts on Lie Groups......Page 758
11.23 Facts on Subspace Decomposition......Page 759
11.24 Notes......Page 765
12.1 State Space and Transfer Function Models......Page 766
12.2 Laplace Transform Analysis......Page 769
12.3 The Unobservable Subspace and Observability......Page 770
12.4 Observable Asymptotic Stability......Page 775
12.5 Detectability......Page 777
12.6 The Controllable Subspace and Controllability......Page 778
12.7 Controllable Asymptotic Stability......Page 786
12.8 Stabilizability......Page 790
12.9 Realization Theory......Page 792
12.10 Zeros......Page 800
12.11 H2 System Norm......Page 808
12.12 Harmonic Steady-State Response......Page 811
12.13 System Interconnections......Page 813
12.14 Standard Control Problem......Page 815
12.15 Linear-Quadratic Control......Page 818
12.16 Solutions of the Riccati Equation......Page 821
12.17 The Stabilizing Solution of the Riccati Equation......Page 825
12.18 The Maximal Solution of the Riccati Equation......Page 830
12.19 Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation......Page 832
12.20 Facts on Stability, Observability, and Controllability......Page 833
12.21 Facts on the Lyapunov Equation and Inertia......Page 836
12.22 Facts on Realizations and the H2 System Norm......Page 841
12.23 Facts on the Riccati Equation......Page 845
12.24 Notes......Page 848
Index......Page 850
Back Cover......Page 1103