# Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein

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1 Matrix Mathematics Theory, Facts, and Formulas with Application to Linear Systems Theory Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

2 Contents Special Symbols xv Conventions, Notation, and Terminology xxvü Preface Acknowledgments xxxiü xxxvü Chapterl. Preliminaries Logic and Sets Relations and Functions Facts on Logic, Sets, and Functions Facts on Scalar Inequalities Notes 12 Chapter2. Basic Matrix Properties Matrix Algebra Transpose and Inner Product Convex Sets, Cones, and Subspaces Range and Null Space Rank and Defect Invertibility Determinants Properties of Partitioned Matrices Facts on Cones, Convex Hulls, and Subspaces Facts on Range, Null Space, Rank, and Defect Facts on Identities Facts on Determinants Facts on Determinants of Partitioned Matrices Facts on Adjugates and Inverses Facts on Inverses of Partitioned Matrices Facts on Commutators Facts on Complex Matrices 75

3 X CONTENTS 2.18 Facts on Geometry Notes 79 Chapter 3. Matrix Classes and Transformations Matrix Classes Matrix Transformations Lie Algebras and Groups Facts on Group-Invertible and Range-Hermitian Matrices Facts on Normal, Hermitian, and Skew-Hermitian Matrices Facts on the Commutator Facts on Unitary Matrices Facts on Idempotent Matrices Facts on Projectors Facts on Reflectors Facts on Nilpotent Matrices Facts on Hamiltonian and Symplectic Matrices Facts on Groups Facts on Quaternions Facts on Miscellaneous Types of Matrices Notes 120 Chapter 4. Matrix Polynomials and Rational Transfer Functions Polynomials Matrix Polynomials The Smith Decomposition and Similarity Invariants Eigenvalues Eigenvectors Minimal Polynomial Rational Transfer Functions and the Smith-McMillan Decomposition Facts on Polynomials Facts on the Characteristic and Minimal Polynomials Facts on the Spectrum Facts on Nonnegative Matrices Notes 160 Chapter 5. Matrix Decompositions Smith Form Multi-Companion Form Hypercompanion Form and Jordan Form Schur Decomposition Eigenstructure Properties Singular Value Decomposition Facts on the Inertia Facts on Matrix Transformations for One Matrix 186

4 CONTENTS XI 5.9 Facts on Matrix Transformations for Two or More Matrices Facts on Eigenvalues and Singular Values for One Matrix Facts on Eigenvalues and Singular Values for Two or More Matrices Facts on Matrix Eigenstructure Facts on Companion, Vandermonde, and Circulant Matrices Facts on Matrix Factorizations Facts on the Polar Decomposition Notes 222 Chapter 6. Generalized Inverses Moore-Penrose Generalized Inverse Drazin Generalized Inverse Facts on the Moore-Penrose Generalized Inverse for One Matrix Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices Facts on the Drazin and Group Generalized Inverses Notes 246 Chapter 7. Kronecker and Schur Algebra Kronecker Product Kronecker Sum and Linear Matrix Equations Schur Product Facts on the Kronecker Product Facts on the Kronecker Sum Facts on the Schur Product Notes 261 Chapter 8. Positive-Semidefinite Matrices Positive-Semidefinite and Positive-Definite Orderings Submatrices Simultaneous Diagonalization Eigenvalue Inequalities Matrix Inequalities Facts on Range and Rank Facts on Identities and Inequalities for One Matrix Facts on Identities and Inequalities for Two or More Matrices Facts on Identities and Inequalities for Partitioned Matrices Facts on the Trace Facts on the Determinant Facts on Quadratic Forms 316

5 XII 8.13 Facts on Matrix Transformations Facts on Eigenvalues and Singular Values Facts on Generalized Inverses Facts on the Kronecker and Schur Products Facts on Majorization Notes 342 Chapter 9. Norms Vector Norms Matrix Norms Compatible Norms Induced Norms Induced Lower Bound Singular Value Inequalities Facts on Vector Norms Facts on Matrix Norms for One Matrix Facts on Matrix Norms for Two or More Matrices Facts on Matrix Norms and Eigenvalues Facts on Singular Values for One Matrix Facts on Singular Values for Two or More Matrices Notes 399 Chapter 10. Functions of Matrices and Their Derivatives Open Sets and Closed Sets Limits Continuity Derivatives Functions of a Matrix Matrix Derivatives Facts on Open, Closed, and Convex Sets Facts on Functions and Derivatives Notes 417 Chapter 11. The Matrix Exponential and Stability Theory Definition of the Matrix Exponential Structure of the Matrix Exponential Explicit Expressions Logarithms Lie Groups Lyapunov Stability Theory Linear Stability Theory The Lyapunov Equation Discrete-Time Stability Theory Facts on Matrix Exponential Formulas Facts on the Matrix Exponential for One Matrix 451

6 CONTENTS XII! Facts on the Matrix Exponential for Two or More Matrices Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for One Matrix Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for Two or More Matrices Facts on Stable Polynomials Facts on Lie Groups Facts on Stable Matrices Facts on Discrete-Time Stability Facts on Subspace Decomposition Notes 485 Chapter 12. Linear Systems and Control Theory State Space and Transfer Function Models Laplace Transform Analysis The Unobservable Subspace and Observability Observable Asymptotic Stability Detectability The Controllable Subspace and Controllability Controllable Asymptotic Stability Stabilizability Realization Theory System Zeros H 2 SystemNorm Harmonie Steady-State Response System Interconnections H 2 Standard Control Problem Linear-Quadratic Control Solutions of the Riccati Equation The Stabilizing Solution of the Riccati Equation The Maximal Solution of the Riccati Equation Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation Facts on Stability, Observability, and Controllability Facts on the Lyapunov Equation and Inertia Facts on Realizations and the H 2 System Norm Facts on the Riccati Equation Notes 561 Bibliography 563 Author Index 611 Index 619

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