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

Preface to the Second Edition......Page 15
Preface to the First Edition......Page 17
Special Symbols......Page 21
Conventions, Notation, and Terminology......Page 33
Logic and Sets......Page 43
Functions......Page 45
Relations......Page 47
Graphs......Page 50
Facts on Logic, Sets, Functions, and Relations......Page 52
Facts on Graphs......Page 55
Facts on Binomial Identities and Sums......Page 56
Facts on Convex Functions......Page 63
Facts on Scalar Identities and Inequalities in One Variable......Page 64
Facts on Scalar Identities and Inequalities in Two Variables......Page 72
Facts on Scalar Identities and Inequalities in Three Variables......Page 81
Facts on Scalar Identities and Inequalities in Four Variables......Page 88
Facts on Scalar Identities and Inequalities in Eight Variables......Page 89
Facts on Scalar Identities and Inequalities in n Variables......Page 90
Facts on Scalar Identities and Inequalities in 2n Variables......Page 102
Facts on Scalar Identities and Inequalities in 3n Variables......Page 109
Facts on Scalar Identities and Inequalities in Complex Variables......Page 110
Facts on Trigonometric and Hyperbolic Identities......Page 116
Notes......Page 118
Matrix Algebra......Page 119
Transpose and Inner Product......Page 126
Convex Sets, Cones, and Subspaces......Page 131
Range and Null Space......Page 135
Rank and Defect......Page 137
Invertibility......Page 140
The Determinant......Page 144
Partitioned Matrices......Page 148
Facts on Polars, Cones, Dual Cones, Convex Hulls, and Subspaces......Page 152
Facts on Range, Null Space, Rank, and Defect......Page 157
Facts on the Range, Rank, Null Space, and Defect ofPartitioned Matrices......Page 162
Facts on the Inner Product, Outer Product, Trace, and Matrix Powers......Page 168
Facts on the Determinant......Page 170
Facts on the Determinant of Partitioned Matrices......Page 174
Facts on Left and Right Inverses......Page 182
Facts on the Adjugate and Inverses......Page 183
Facts on the Inverse of Partitioned Matrices......Page 188
Facts on Commutators......Page 191
Facts on Complex Matrices......Page 193
Facts on Geometry......Page 196
Facts on Majorization......Page 204
Notes......Page 206
Matrix Classes......Page 207
Matrices Based on Graphs......Page 212
Lie Algebras and Groups......Page 213
Matrix Transformations......Page 215
Projectors, Idempotent Matrices, and Subspaces......Page 217
Facts on Group-Invertible and Range-Hermitian Matrices......Page 219
Facts on Normal, Hermitian, and Skew-Hermitian Matrices......Page 220
Facts on Commutators......Page 226
Facts on Linear Interpolation......Page 227
Facts on the Cross Product......Page 228
Facts on Unitary and Shifted-Unitary Matrices......Page 231
Facts on Idempotent Matrices......Page 240
Facts on Projectors......Page 248
Facts on Reflectors......Page 253
Facts on Tripotent Matrices......Page 254
Facts on Nilpotent Matrices......Page 255
Facts on Hankel and Toeplitz Matrices......Page 257
Facts on Hamiltonian and Symplectic Matrices......Page 258
Facts on Miscellaneous Types of Matrices......Page 259
Facts on Groups......Page 263
Facts on Quaternions......Page 267
Notes......Page 271
Polynomials......Page 273
Polynomial Matrices......Page 276
The Smith Decomposition and Similarity Invariants......Page 278
Eigenvalues......Page 281
Eigenvectors......Page 287
The Minimal Polynomial......Page 289
Rational Transfer Functions and the Smith-McMillanDecomposition......Page 291
Facts on Polynomials and Rational Functions......Page 295
Facts on the Characteristic and Minimal Polynomials......Page 302
Facts on the Spectrum......Page 307
Facts on Graphs and Nonnegative Matrices......Page 314
Notes......Page 323
Multicompanion Form......Page 325
Hypercompanion Form and Jordan Form......Page 329
Schur Decomposition......Page 334
Eigenstructure Properties......Page 337
Singular Value Decomposition......Page 343
Pencils and the Kronecker Canonical Form......Page 346
Facts on the Inertia......Page 349
Facts on Matrix Transformations for One Matrix......Page 353
Facts on Matrix Transformations for Two or More Matrices......Page 358
Facts on Eigenvalues and Singular Values for One Matrix......Page 363
Facts on Eigenvalues and Singular Values for Two or MoreMatrices......Page 375
Facts on Matrix Eigenstructure......Page 380
Facts on Matrix Factorizations......Page 387
Facts on Companion, Vandermonde, and Circulant Matrices......Page 394
Facts on Simultaneous Transformations......Page 400
Facts on the Polar Decomposition......Page 401
Facts on Additive Decompositions......Page 402
Notes......Page 403
Moore-Penrose Generalized Inverse......Page 405
Drazin Generalized Inverse......Page 409
Facts on the Moore-Penrose Generalized Inverse for OneMatrix......Page 411
Facts on the Moore-Penrose Generalized Inverse for Two orMore Matrices......Page 419
Facts on the Moore-Penrose Generalized Inverse forPartitioned Matrices......Page 427
Facts on the Drazin and Group Generalized Inverses......Page 435
Notes......Page 440
Kronecker Product......Page 441
Kronecker Sum and Linear Matrix Equations......Page 444
Schur Product......Page 446
Facts on the Kronecker Product......Page 447
Facts on the Kronecker Sum......Page 451
Facts on the Schur Product......Page 455
Notes......Page 458
Positive-Semidefinite and Positive-Definite Orderings......Page 459
Submatrices......Page 461
Simultaneous Diagonalization......Page 464
Eigenvalue Inequalities......Page 466
Exponential, Square Root, and Logarithm of Hermitian Matrices......Page 472
Matrix Inequalities......Page 473
Facts on Range and Rank......Page 485
Facts on Structured Positive-Semidefinite Matrices......Page 486
Facts on Identities and Inequalities for One Matrix......Page 492
Facts on Identities and Inequalities for Two or More Matrices......Page 498
Facts on Identities and Inequalities for Partitioned Matrices......Page 509
Facts on the Trace......Page 517
Facts on the Determinant......Page 527
Facts on Convex Sets and Convex Functions......Page 536
Facts on Quadratic Forms......Page 542
Facts on Simultaneous Diagonalization......Page 549
Facts on Eigenvalues and Singular Values for One Matrix......Page 550
Facts on Eigenvalues and Singular Values for Two or MoreMatrices......Page 554
Facts on Alternative Partial Orderings......Page 564
Facts on Generalized Inverses......Page 567
Facts on the Kronecker and Schur Products......Page 573
Notes......Page 583
Vector Norms......Page 585
Matrix Norms......Page 588
Compatible Norms......Page 591
Induced Norms......Page 595
Induced Lower Bound......Page 600
Singular Value Inequalities......Page 602
Facts on Vector Norms......Page 605
Facts on Matrix Norms for One Matrix......Page 613
Facts on Matrix Norms for Two or More Matrices......Page 622
Facts on Matrix Norms for Partitioned Matrices......Page 635
Facts on Matrix Norms and Eigenvalues Involving One Matrix......Page 638
Facts on Matrix Norms and Eigenvalues Involving Two or More Matrices......Page 641
Facts on Matrix Norms and Singular Values for One Matrix......Page 644
Facts on Matrix Norms and Singular Values for Two or More Matrices......Page 649
Facts on Least Squares......Page 660
Notes......Page 661
Open Sets and Closed Sets......Page 663
Limits......Page 664
Continuity......Page 665
Derivatives......Page 667
Functions of a Matrix......Page 670
Matrix Square Root and Matrix Sign Functions......Page 671
Matrix Derivatives......Page 672
Facts Involving One Set......Page 674
Facts Involving Two or More Sets......Page 676
Facts on Matrix Functions......Page 679
Facts on Functions and Derivatives......Page 680
Notes......Page 684
Definition of the Matrix Exponential......Page 685
Structure of the Matrix Exponential......Page 688
Explicit Expressions......Page 693
Matrix Logarithms......Page 696
The Logarithm Function......Page 698
Lie Groups......Page 700
Lyapunov Stability Theory......Page 702
Linear Stability Theory......Page 704
The Lyapunov Equation......Page 708
Discrete-Time Stability Theory......Page 711
Facts on Matrix Exponential Formulas......Page 713
Facts on the Matrix Exponential for One Matrix......Page 719
Facts on the Matrix Exponential for Two or More Matrices......Page 723
Facts on the Matrix Exponential and Eigenvalues,Singular Values, and Norms for One Matrix......Page 731
Facts on the Matrix Exponential and Eigenvalues,Singular Values, and Norms for Two or More Matrices......Page 734
Facts on Stable Polynomials......Page 737
Facts on Stable Matrices......Page 740
Facts on Almost Nonnegative Matrices......Page 748
Facts on Discrete-Time-Stable Polynomials......Page 750
Facts on Discrete-Time-Stable Matrices......Page 754
Facts on Lie Groups......Page 757
Facts on Subspace Decomposition......Page 758
Notes......Page 764
State Space and Transfer Function Models......Page 765
Laplace Transform Analysis......Page 768
The Unobservable Subspace and Observability......Page 769
Observable Asymptotic Stability......Page 774
Detectability......Page 776
The Controllable Subspace and Controllability......Page 777
Controllable Asymptotic Stability......Page 785
Stabilizability......Page 789
Realization Theory......Page 791
Zeros......Page 799
H2-2mu System Norm......Page 807
Harmonic Steady-State Response......Page 810
System Interconnections......Page 812
Standard Control Problem......Page 814
Linear-Quadratic Control......Page 817
Solutions of the Riccati Equation......Page 820
The Stabilizing Solution of the Riccati Equation......Page 824
The Maximal Solution of the Riccati Equation......Page 829
Positive-Semidefinite and Positive-Definite Solutions of theRiccati Equation......Page 831
Facts on Stability, Observability, and Controllability......Page 832
Facts on the Lyapunov Equation and Inertia......Page 835
Facts on Realizations and the H2-2mu System Norm......Page 840
Facts on the Riccati Equation......Page 844
Notes......Page 847
Index......Page 849