Matrix Mathematics

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Preface to the Second Edition xv Preface to the First Edition xvii Special Symbols xxi Conventions, Notation, and Terminology xxxiii Chapter 1: Preliminaries 1 1.1 Logic 1 1.2 Sets 2 1.3 Integers, Real Numbers, and Complex Numbers 3 1.4 Functions 4 1.5 Relations 6 1.6 Graphs 9 1.7 Facts on Logic, Sets, Functions, and Relations 11 1.8 Facts on Graphs 15 1.9 Facts on Binomial Identities and Sums 16 1.10 Facts on Convex Functions 23 1.11 Facts on Scalar Identities and Inequalities in One Variable 25 1.12 Facts on Scalar Identities and Inequalities in Two Variables 33 1.13 Facts on Scalar Identities and Inequalities in Three Variables 42 1.14 Facts on Scalar Identities and Inequalities in Four Variables 50 1.15 Facts on Scalar Identities and Inequalities in Six Variables 52 1.16 Facts on Scalar Identities and Inequalities in Eight Variables 52 1.17 Facts on Scalar Identities and Inequalities in n Variables 52 1.18 Facts on Scalar Identities and Inequalities in 2n Variables 66 1.19 Facts on Scalar Identities and Inequalities in 3n Variables 74 1.20 Facts on Scalar Identities and Inequalities in Complex Variables 74 1.21 Facts on Trigonometric and Hyperbolic Identities 81 1.22 Notes 84 Chapter 2: Basic Matrix Properties 85 2.1 Matrix Algebra 85 2.2 Transpose and Inner Product 92 2.3 Convex Sets, Cones, and Subspaces 97 2.4 Range and Null Space 101 2.5 Rank and Defect 104 2.6 Invertibility 106 2.7 The Determinant 111 2.8 Partitioned Matrices 115 2.9 Facts on Polars, Cones, Dual Cones, Convex Hulls, and Subspaces 119 2.10 Facts on Range, Null Space, Rank, and Defect 124 2.11 Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices 130 2.12 Facts on the Inner Product, Outer Product, Trace, and Matrix Powers 136 2.13 Facts on the Determinant 139 2.14 Facts on the Determinant of Partitioned Matrices 144 2.15 Facts on Left and Right Inverses 152 2.16 Facts on the Adjugate and Inverses 153 2.17 Facts on the Inverse of Partitioned Matrices 159 2.18 Facts on Commutators 161 2.19 Facts on Complex Matrices 164 2.20 Facts on Geometry 167 2.21 Facts on Majorization 175 2.22 Notes 178 Chapter 3: Matrix Classes and Transformations 179 3.1 Matrix Classes 179 3.2 Matrices Related to Graphs 184 3.3 Lie Algebras and Groups 185 3.4 Matrix Transformations 188 3.5 Projectors, Idempotent Matrices, and Subspaces 190 3.6 Facts on Group-Invertible and Range-Hermitian Matrices 191 3.7 Facts on Normal, Hermitian, and Skew-Hermitian Matrices 192 3.8 Facts on Commutators 199 3.9 Facts on Linear Interpolation 200 3.10 Facts on the Cross Product 202 3.11 Facts on Unitary and Shifted-Unitary Matrices 205 3.12 Facts on Idempotent Matrices 215 3.13 Facts on Projectors 223 3.14 Facts on Reflectors 229 3.15 Facts on Involutory Matrices 230 3.16 Facts on Tripotent Matrices 231 3.17 Facts on Nilpotent Matrices 232 3.18 Facts on Hankel and Toeplitz Matrices 234 3.19 Facts on Tridiagonal Matrices 237 3.20 Facts on Hamiltonian and Symplectic Matrices 238 3.21 Facts on Matrices Related to Graphs 240 3.22 Facts on Triangular, Irreducible, Cauchy, Dissipative, Contractive, and Centrosymmetric Matrices 240 3.23 Facts on Groups 242 3.24 Facts on Quaternions 247 3.25 Notes 252 Chapter 4: Polynomial Matrices and Rational Transfer Functions 253 4.1 Polynomials 253 4.2 Polynomial Matrices 256 4.3 The Smith Decomposition and Similarity Invariants 258 4.4 Eigenvalues 261 4.5 Eigenvectors 267 4.6 The Minimal Polynomial 269 4.7 Rational Transfer Functions and the Smith-McMillan Decomposition 271 4.8 Facts on Polynomials and Rational Functions 276 4.9 Facts on the Characteristic and Minimal Polynomials 282 4.10 Facts on the Spectrum 288 4.11 Facts on Graphs and Nonnegative Matrices 297 4.12 Notes 307 Chapter 5: Matrix Decompositions 309 5.1 Smith Form 309 5.2 Multicompanion Form 309 5.3 Hypercompanion Form and Jordan Form 314 5.4 Schur Decomposition 318 5.5 Eigenstructure Properties 321 5.6 Singular Value Decomposition 328 5.7 Pencils and the Kronecker Canonical Form 330 5.8 Facts on the Inertia 334 5.9 Facts on Matrix Transformations for One Matrix 338 5.10 Facts on Matrix Transformations for Two or More Matrices 345 5.11 Facts on Eigenvalues and Singular Values for One Matrix 350 5.12 Facts on Eigenvalues and Singular Values for Two or More Matrices 362 5.13 Facts on Matrix Pencils 369 5.14 Facts on Matrix Eigenstructure 369 5.15 Facts on Matrix Factorizations 377 5.16 Facts on Companion, Vandermonde, Circulant, and Hadamard Matrices 385 5.17 Facts on Simultaneous Transformations 391 5.18 Facts on the Polar Decomposition 393 5.19 Facts on Additive Decompositions 394 5.20 Notes 396 Chapter 6: Generalized Inverses 397 6.1 Moore-Penrose Generalized Inverse 397 6.2 Drazin Generalized Inverse 401 6.3 Facts on the Moore-Penrose Generalized Inverse for One Matrix 404 6.4 Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices 411 6.5 Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices 422 6.6 Facts on the Drazin and Group Generalized Inverses 431 6.7 Notes 438 Chapter 7: Kronecker and Schur Algebra 439 7.1 Kronecker Product 439 7.2 Kronecker Sum and Linear Matrix Equations 443 7.3 Schur Product 444 7.4 Facts on the Kronecker Product 445 7.5 Facts on the Kronecker Sum 450 7.6 Facts on the Schur Product 454 7.7 Notes 458 Chapter 8: Positive-Semidefinite Matrices 459 8.1 Positive-Semidefinite and Positive-Definite Orderings 459 8.2 Submatrices 461 8.3 Simultaneous Diagonalization 465 8.4 Eigenvalue Inequalities 467 8.5 Exponential, Square Root, and Logarithm of Hermitian Matrices 473 8.6 Matrix Inequalities 474 8.7 Facts on Range and Rank 486 8.8 Facts on Structured Positive-Semidefinite Matrices 488 8.9 Facts on Identities and Inequalities for One Matrix 495 8.10 Facts on Identities and Inequalities for Two or More Matrices 501 8.11 Facts on Identities and Inequalities for Partitioned Matrices 514 8.12 Facts on the Trace 523 8.13 Facts on the Determinant 533 8.14 Facts on Convex Sets and Convex Functions 543 8.15 Facts on Quadratic Forms 550 8.16 Facts on the Gaussian Density 556 8.17 Facts on Simultaneous Diagonalization 558 8.18 Facts on Eigenvalues and Singular Values for One Matrix 559 8.19 Facts on Eigenvalues and Singular Values for Two or More Matrices 564 8.20 Facts on Alternative Partial Orderings 574 8.21 Facts on Generalized Inverses 577 8.22 Facts on the Kronecker and Schur Products 584 8.23 Notes 595 Chapter 9: Norms 597 9.1 Vector Norms 597 9.2 Matrix Norms 601 9.3 Compatible Norms 604 9.4 Induced Norms 607 9.5 Induced Lower Bound 613 9.6 Singular Value Inequalities 615 9.7 Facts on Vector Norms 618 9.8 Facts on Matrix Norms for One Matrix 627 9.9 Facts on Matrix Norms for Two or More Matrices 636 9.10 Facts on Matrix Norms for Partitioned Matrices 649 9.11 Facts on Matrix Norms and Eigenvalues for One Matrix 653 9.12 Facts on Matrix Norms and Eigenvalues for Two or More Matrices 656 9.13 Facts on Matrix Norms and Singular Values for One Matrix 659 9.14 Facts on Matrix Norms and Singular Values for Two or More Matrices 665 9.15 Facts on Linear Equations and Least Squares 676 9.16 Notes 680 Chapter 10: Functions of Matrices and Their Derivatives 681 10.1 Open Sets and Closed Sets 681 10.2 Limits 682 10.3 Continuity 684 10.4 Derivatives 685 10.5 Functions of a Matrix 688 10.6 Matrix Square Root and Matrix Sign Functions 690 10.7 Matrix Derivatives 690 10.8 Facts on One Set 693 10.9 Facts on Two or More Sets 695 10.10 Facts on Matrix Functions 698 10.11 Facts on Functions 699 10.12 Facts on Derivatives 701 10.13 Facts on Infinite Series 704 10.14 Notes 705 Chapter 11: The Matrix Exponential and Stability Theory 707 11.1 Definition of the Matrix Exponential 707 11.2 Structure of the Matrix Exponential 710 11.3 Explicit Expressions 715 11.4 Matrix Logarithms 718 11.5 Principal Logarithm 720 11.6 Lie Groups 722 11.7 Lyapunov Stability Theory 725 11.8 Linear Stability Theory 726 11.9 The Lyapunov Equation 730 11.10 Discrete-Time Stability Theory 734 11.11 Facts on Matrix Exponential Formulas 736 11.12 Facts on the Matrix Sine and Cosine 742 11.13 Facts on the Matrix Exponential for One Matrix 743 11.14 Facts on the Matrix Exponential for Two or More Matrices 746 11.15 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for One Matrix 756 11.16 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for Two or More Matrices 759 11.17 Facts on Stable Polynomials 763 11.18 Facts on Stable Matrices 766 11.19 Facts on Almost Nonnegative Matrices 774 11.20 Facts on Discrete-Time-Stable Polynomials 777 11.21 Facts on Discrete-Time-Stable Matrices 782 11.22 Facts on Lie Groups 786 11.23 Facts on Subspace Decomposition 786 11.24 Notes 793 Chapter 12: Linear Systems and Control Theory 795 12.1 State Space and Transfer Function Models 795 12.2 Laplace Transform Analysis 798 12.3 The Unobservable Subspace and Observability 800 12.4 Observable Asymptotic Stability 805 12.5 Detectability 807 12.6 The Controllable Subspace and Controllability 808 12.7 Controllable Asymptotic Stability 816 12.8 Stabilizability 820 12.9 Realization Theory 822 12.10 Zeros 830 12.11 H2 System Norm 838 12.12 Harmonic Steady-State Response 841 12.13 System Interconnections 842 12.14 Standard Control Problem 845 12.15 Linear-Quadratic Control 847 12.16 Solutions of the Riccati Equation 850 12.17 The Stabilizing Solution of the Riccati Equation 855 12.18 The Maximal Solution of the Riccati Equation 859 12.19 Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation 862 12.20 Facts on Stability, Observability, and Controllability 863 12.21 Facts on the Lyapunov Equation and Inertia 866 12.22 Facts on Realizations and the H2 System Norm 872 12.23 Facts on the Riccati Equation 875 12.24 Notes 879 Bibliography 881 Author Index 967 Index 979

Dennis S. Bernstein is professor of aerospace engineering at the University of Michigan.

"When a matrix question is thrown my way, I will now refer my correspondents ... to Bernstein's handbook."--Philip J. Davis, SIAM News "Matrix Mathematics contains an impressive collection of definitions, relations, properties, equations, inequalities, and facts centered around matrices and their use in systems and control. The amount of material that is covered is quite impressive and well structured... I highly recommend the book as a source for retrieving matrix results that one would otherwise have to search for in the extensive literature on matrix theory."--Paul Van Dooren, IEEE Control Systems Magazine "The author was very successful in collecting the enormous amount of results in matrix theory in a single source... A beautiful work and an admirable performance!"--Monatshefte fur Mathematik "It is a remarkable source of matrix results. I will put it on the shelf near to my desk so that I have quick access to it. The book is an impressive accomplishment by the author... I can enthusiastically recommend it to anyone who uses matrices. The author has to be applauded for the accomplishment of putting together this impressive volume."--Helmut Lutkepohl, Image "The book is a well-organized treasure trove of information for anyone interested in matrices and their applications. Look through the Table of Contents and see if there isn't some section that will tempt you and/or illuminate your pathway through the extensive literature on matrix theory. Researchers should have access to this authoritative and comprehensive volume. Academic and industrial libraries should have it in their reference collections. Their patrons will be grateful."--Henry Ricardo, MAA Reviews "Anybody, regardless of level of expertise, could learn new things just by browsing. Open to a random page and start reading. The reader who seeks specific information to solve a problem may also find success here."--David S. Watkins, SIAM Review

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