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```Contents
Preface to the Second Edition xv
Preface to the First Edition xvii
Special Symbols xxi
Conventions, Notation, and Terminology xxxiii
1. Preliminaries 1
1.1 Logic and Sets 1
1.2 Functions 3
1.3 Relations 5
1.4 Graphs 7
1.5 Facts on Logic, Sets, Functions, and Relations 10
1.6 Facts on Graphs 13
1.7 Facts on Binomial Identities and Sums 14
1.8 Facts on Convex Functions 19
1.9 Facts on Scalar Identities and Inequalities in One Variable 20
1.10 Facts on Scalar Identities and Inequalities in Two Variables 28
1.11 Facts on Scalar Identities and Inequalities in Three Variables 36
1.12 Facts on Scalar Identities and Inequalities in Four Variables 44
1.13 Facts on Scalar Identities and Inequalities in Six Variables 45
1.14 Facts on Scalar Identities and Inequalities in Eight Variables 45
1.15 Facts on Scalar Identities and Inequalities in n Variables 46
1.16 Facts on Scalar Identities and Inequalities in 2n Variables 57
1.17 Facts on Scalar Identities and Inequalities in 3n Variables 65
1.18 Facts on Scalar Identities and Inequalities in Complex Variables 65
1.19 Facts on Trigonometric and Hyperbolic Identities 71
1.20 Notes 73
2. Basic Matrix Properties 75
2.1 Matrix Algebra 75
2.2 Transpose and Inner Product 82
2.3 Convex Sets, Cones, and Subspaces 87
2.4 Range and Null Space 91
2.5 Rank and Defect 93
2.6 Invertibility 96
2.7 The Determinant 100
2.8 Partitioned Matrices 104
2.9 Facts on Polars, Cones, Dual Cones, Convex Hulls, and Sub-spaces 108
2.10 Facts on Range, Null Space, Rank, and Defect 112
2.11 Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices 117
2.12 Facts on the Trace, Outer Product, and Matrix Powers 122
2.13 Facts on the Determinant 125
2.14 Facts on the Determinant of Partitioned Matrices 129
2.15 Facts on Left and Right Inverses 136
2.16 Facts on the Adjugate and Inverses 137
2.17 Facts on the Inverse of Partitioned Matrices 143
2.18 Facts on Commutators 146
2.19 Facts on Complex Matrices 148
2.20 Facts on Geometry 151
2.21 Facts on Majorization 157
2.22 Notes 159
3. Matrix Classes and Transformations 161
3.1 Matrix Classes 161
3.2 Matrices Based on Graphs 166
3.3 Lie Algebras and Groups 167
3.4 Matrix Transformations 169
3.5 Projectors, Idempotent Matrices, and Subspaces 171
3.6 Facts on Group-Invertible and Range-Hermitian Matrices 173
3.7 Facts on Normal, Hermitian, and Skew-Hermitian Matrices 174
3.8 Facts on Commutators 180
3.9 Facts on Linear Interpolation 181
3.10 Facts on the Cross Product 182
3.11 Facts on Unitary and Shifted-Unitary Matrices 184
3.12 Facts on Idempotent Matrices 192
3.13 Facts on Projectors 200
3.14 Facts on Reflectors 204
3.15 Facts on Involutory, Skew-Involutory, and Tripotent Matrices 205
3.16 Facts on Nilpotent Matrices 206
3.17 Facts on Hankel and Toeplitz Matrices 208
3.18 Facts on Hamiltonian and Symplectic Matrices 209
3.19 Facts on Miscellaneous Types of Matrices 211
3.20 Facts on Groups 213
3.21 Facts on Quaternions 217
3.22 Notes 221
4. Polynomial Matrices and Rational Transfer Functions 223
4.1 Polynomials 223
4.2 Polynomial Matrices 226
4.3 The Smith Decomposition and Similarity Invariants 228
4.4 Eigenvalues 231
4.5 Eigenvectors 237
4.6 The Minimal Polynomial 239
4.7 Rational Transfer Functions and the Smith-McMillan Decomposition 241
4.8 Facts on Polynomials and Rational Functions 245
4.9 Facts on the Characteristic and Minimal Polynomials 252
4.10 Facts on the Spectrum 257
4.11 Facts on Graphs and Nonnegative Matrices 264
4.12 Notes 272
5. Matrix Decompositions 273
5.1 Smith Form 273
5.2 Multicompanion Form 273
5.3 Hypercompanion Form and Jordan Form 277
5.4 Schur Decomposition 282
5.5 Eigenstructure Properties 285
5.6 Singular Value Decomposition 290
5.7 Pencils and the Kronecker Canonical Form 293
5.8 Facts on the Inertia 297
5.9 Facts on Matrix Transformations for One Matrix 301
5.10 Facts on Matrix Transformations for Two or More Matrices 306
5.11 Facts on Eigenvalues and Singular Values for One Matrix 310
5.12 Facts on Eigenvalues and Singular Values for Two or More Matrices 322
5.13 Facts on Matrix Pencils 328
5.14 Facts on Matrix Eigenstructure 328
5.15 Facts on Matrix Factorizations 335
5.16 Facts on Companion, Vandermonde, and Circulant Matrices 341
5.17 Facts on Simultaneous Transformations 347
5.18 Facts on the Polar Decomposition 348
5.19 Facts on Additive Decompositions 349
5.20 Notes 350
6. Generalized Inverses 353
6.1 Moore-Penrose Generalized Inverse 353
6.2 Drazin Generalized Inverse 357
6.3 Facts on the Moore-Penrose Generalized Inverse for One Matrix 359
6.4 Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices 366
6.5 Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices 374
6.6 Facts on the Drazin and Group Generalized Inverses 382
6.7 Notes 387
7. Kronecker and Schur Algebra 389
7.1 Kronecker Product 389
7.2 Kronecker Sum and Linear Matrix Equations 392
7.3 Schur Product 394
7.4 Facts on the Kronecker Product 394
7.5 Facts on the Kronecker Sum 398
7.6 Facts on the Schur Product 402
7.7 Notes 406
8. Positive-Semidefinite Matrices 407
8.1 Positive-Semidefinite and Positive-Definite Orderings 407
8.2 Submatrices 409
8.3 Simultaneous Diagonalization 412
8.4 Eigenvalue Inequalities 414
8.5 Exponential and Logarithm of Hermitian Matrices 420
8.6 Matrix Inequalities 421
8.7 Facts on Range and Rank 432
8.8 Facts on Structured Positive-Semidefinite Matrices 434
8.9 Facts on Identities and Inequalities for One Matrix 440
8.10 Facts on Identities and Inequalities for Two or More Matrices 445
8.11 Facts on Identities and Inequalities for Partitioned Matrices 456
8.12 Facts on the Trace 464
8.13 Facts on the Determinant 473
8.14 Facts on Convex Sets and Convex Functions 482
8.15 Facts on Quadratic Forms 487
8.16 Facts on Simultaneous Diagonalization 494
8.17 Facts on Eigenvalues and Singular Values for One Matrix 495
8.18 Facts on Eigenvalues and Singular Values for Two or More Matrices 500
8.19 Facts on Alternative Partial Orderings 509
8.20 Facts on Generalized Inverses 511
8.21 Facts on the Kronecker and Schur Products 518
8.22 Notes 528
9. Norms 529
9.1 Vector Norms 529
9.2 Matrix Norms 532
9.3 Compatible Norms 535
9.4 Induced Norms 539
9.5 Induced Lower Bound 544
9.6 Singular Value Inequalities 546
9.7 Facts on Vector Norms 549
9.8 Facts on Matrix Norms for One Matrix 557
9.9 Facts on Matrix Norms for Two or More Matrices 565
9.10 Facts on Matrix Norms for Partitioned Matrices 578
9.11 Facts on Matrix Norms and Eigenvalues Involving One Matrix 582
9.12 Facts on Matrix Norms and Eigenvalues Involving Two or More Matrices 584
9.13 Facts on Matrix Norms and Singular Values for One Matrix 586
9.14 Facts on Matrix Norms and Singular Values for Two or More Matrices 591
9.15 Facts on Least Squares 602
9.16 Notes 603
10.Functions of Matrices and Their Derivatives 605
10.1 Open Sets and Closed Sets 605
10.2 Limits 606
10.3 Continuity 607
10.4 Derivatives 609
10.5 Functions of a Matrix 612
10.6 Matrix Derivatives 613
10.7 Facts Involving One Set 615
10.8 Facts Involving Two or More Sets 618
10.9 Facts on Functions and Derivatives 620
10.10 Notes 625
11.The Matrix Exponential and Stability Theory 627
11.1 Definition of the Matrix Exponential 627
11.2 Structure of the Matrix Exponential 630
11.3 Explicit Expressions 635
11.4 Matrix Logarithms 638
11.5 The Logarithm Function 640
11.6 Lie Groups 641
11.7 Lyapunov Stability Theory 643
11.8 Linear Stability Theory 645
11.9 The Lyapunov Equation 649
11.10 Discrete-Time Stability Theory 652
11.11 Facts on Matrix Exponential Formulas 654
11.12 Facts on the Matrix Exponential for One Matrix 659
11.13 Facts on the Matrix Exponential for Two or More Matrices 663
11.14 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for
One Matrix 670
11.15 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for
Two or More Matrices 673
11.16 Facts on Stable Polynomials 676
11.17 Facts on Stable Matrices 679
11.18 Facts on Almost Nonnegative Matrices 686
11.19 Facts on Discrete-Time-Stable Polynomials 688
11.20 Facts on Discrete-Time-Stable Matrices 692
11.21 Facts on Lie Groups 695
11.22 Facts on Subspace Decomposition 696
11.23 Notes 702
12.Linear Systems and Control Theory 703
12.1 State Space and Transfer Function Models 703
12.2 Laplace Transform Analysis 706
12.3 The Unobservable Subspace and Observability 707
12.4 Observable Asymptotic Stability 712
12.5 Detectability 714
12.6 The Controllable Subspace and Controllability 715
12.7 Controllable Asymptotic Stability 723
12.8 Stabilizability 727
12.9 Realization Theory 729
12.10 Zeros 737
12.11 H2 System Norm 745
12.13 System Interconnections 750
12.14 Standard Control Problem 753
12.16 Solutions of the Riccati Equation 758
12.17 The Stabilizing Solution of the Riccati Equation 762
12.18 The Maximal Solution of the Riccati Equation 767
12.19 Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation 769
12.20 Facts on Stability, Observability, and Controllability 770
12.21 Facts on the Lyapunov Equation and Inertia 773
12.22 Facts on Realizations and the H2 System Norm 778
12.23 Facts on the Riccati Equation 782
12.24 Notes 785
Bibliography 787
Author Index 869
Index 881
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Library of Congress Subject Headings for this publication:

Matrices.
Linear systems.