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Matrix Mathematics - Theory, Facts, and Formulas, Second Edition.pdf

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    MATRIX MATHEMATICS: THEORY, FACTS, AND FORMULAS, 2ND ED.
    Title Page
    Copyright Page
    Dedication
    Preface to the Second Edition
    Preface to the First Edition
    Special Symbols
    Conventions, Notation, and Terminology
    Chapter 1. Preliminaries
    1.1 Logic and Sets
    1.2 Functions
    1.3 Relations
    1.4 Graphs
    1.5 Facts on Logic, Sets, Functions, and Relations
    1.6 Facts on Graphs
    1.7 Facts on Binomial Identities and Sums
    1.8 Facts on Convex Functions
    1.9 Facts on Scalar Identities and Inequalities in One Variable
    1.10 Facts on Scalar Identities and Inequalities in Two Variables
    1.11 Facts on Scalar Identities and Inequalities in Three Variables
    1.12 Facts on Scalar Identities and Inequalities in Four Variables
    1.13 Facts on Scalar Identities and Inequalities in Six Variables
    1.14 Facts on Scalar Identities and Inequalities in Eight Variables
    1.15 Facts on Scalar Identities and Inequalities in n Variables
    1.16 Facts on Scalar Identities and Inequalities in 2n Variables
    1.17 Facts on Scalar Identities and Inequalities in 3n Variables
    1.18 Facts on Scalar Identities and Inequalities in Complex Variables
    1.19 Facts on Trigonometric and Hyperbolic Identities
    1.20 Notes
    Chapter 2. Basic Matrix Properties
    2.1 Matrix Algebra
    2.2 Transpose and Inner Product
    2.3 Convex Sets, Cones, and Subspaces
    2.4 Range and Null Space
    2.5 Rank and Defect
    2.6 Invertibility
    2.7 The Determinant
    2.8 Partitioned Matrices
    2.9 Facts on Polars, Cones, Dual Cones, Convex Hulls, and Subspaces
    2.10 Facts on Range, Null Space, Rank, and Defect
    2.11 Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices
    2.12 Facts on the Inner Product, Outer Product, Trace, and Matrix Powers
    2.13 Facts on the Determinant
    2.14 Facts on the Determinant of Partitioned Matrices
    2.15 Facts on Left and Right Inverses
    2.16 Facts on the Adjugate and Inverses
    2.17 Facts on the Inverse of Partitioned Matrices
    2.18 Facts on Commutators
    2.19 Facts on Complex Matrices
    2.20 Facts on Geometry
    2.21 Facts on Majorization
    2.22 Notes
    Chapter 3. Matrix Classes and Transformations
    3.1 Matrix Classes
    3.2 Matrices Based on Graphs
    3.3 Lie Algebras and Groups
    3.4 Matrix Transformations
    3.5 Projectors, Idempotent Matrices, and Subspaces
    3.6 Facts on Group-Invertible and Range-Hermitian Matrices
    3.7 Facts on Normal, Hermitian, and Skew-Hermitian Matrices
    3.8 Facts on Commutators
    3.9 Facts on Linear Interpolation
    3.10 Facts on the Cross Product
    3.11 Facts on Unitary and Shifted-Unitary Matrices
    3.12 Facts on Idempotent Matrices
    3.13 Facts on Projectors
    3.14 Facts on Reflectors
    3.15 Facts on Involutory Matrices
    3.16 Facts on Tripotent Matrices
    3.17 Facts on Nilpotent Matrices
    3.18 Facts on Hankel and Toeplitz Matrices
    3.19 Facts on Hamiltonian and Symplectic Matrices
    3.20 Facts on Miscellaneous Types of Matrices
    3.21 Facts on Groups
    3.22 Facts on Quaternions
    3.23 Notes
    Chapter 4. Polynomial Matrices and Rational Transfer Functions
    4.1 Polynomials
    4.2 Polynomial Matrices
    4.3 The Smith Decomposition and Similarity Invariants
    4.4 Eigenvalues
    4.5 Eigenvectors
    4.6 The Minimal Polynomial
    4.7 Rational Transfer Functions and the Smith-McMillan Decomposition
    4.8 Facts on Polynomials and Rational Functions
    4.9 Facts on the Characteristic and Minimal Polynomials
    4.10 Facts on the Spectrum
    4.11 Facts on Graphs and Nonnegative Matrices
    4.12 Notes
    Chapter 5. Matrix Decompositions
    5.1 Smith Form
    5.2 Multicompanion Form
    5.3 Hypercompanion Form and Jordan Form
    5.4 Schur Decomposition
    5.5 Eigenstructure Properties
    5.6 Singular Value Decomposition
    5.7 Pencils and the Kronecker Canonical Form
    5.8 Facts on the Inertia
    5.9 Facts on Matrix Transformations for One Matrix
    5.10 Facts on Matrix Transformations for Two or More Matrices
    5.11 Facts on Eigenvalues and Singular Values for One Matrix
    5.12 Facts on Eigenvalues and Singular Values for Two or More Matrices
    5.13 Facts on Matrix Pencils
    5.14 Facts on Matrix Eigenstructure
    5.15 Facts on Matrix Factorizations
    5.16 Facts on Companion, Vandermonde, and Circulant Matrices
    5.17 Facts on Simultaneous Transformations
    5.18 Facts on the Polar Decomposition
    5.19 Facts on Additive Decompositions
    5.20 Notes
    Chapter 6. Generalized Inverses
    6.1 Moore-Penrose Generalized Inverse
    6.2 Drazin Generalized Inverse
    6.3 Facts on the Moore-Penrose Generalized Inverse for One Matrix
    6.4 Facts on the Moore-Penrose Generalized Inverse for Two or More Matrices
    6.5 Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices
    6.6 Facts on the Drazin and Group Generalized Inverses
    6.7 Notes
    Chapter 7. Kronecker and Schur Algebra
    7.1 Kronecker Product
    7.2 Kronecker Sum and Linear Matrix Equations
    7.3 Schur Product
    7.4 Facts on the Kronecker Product
    7.5 Facts on the Kronecker Sum
    7.6 Facts on the Schur Product
    7.7 Notes
    Chapter 8. Positive-Semidefinite Matrices
    8.1 Positive-Semidefinite and Positive-Definite Orderings
    8.2 Submatrices
    8.3 Simultaneous Diagonalization
    8.4 Eigenvalue Inequalities
    8.5 Exponential, Square Root, and Logarithm of Hermitian Matrices
    8.6 Matrix Inequalities
    8.7 Facts on Range and Rank
    8.8 Facts on Structured Positive-Semidefinite Matrices
    8.9 Facts on Identities and Inequalities for One Matrix
    8.10 Facts on Identities and Inequalities for Two or More Matrices
    8.11 Facts on Identities and Inequalities for Partitioned Matrices
    8.12 Facts on the Trace
    8.13 Facts on the Determinant
    8.14 Facts on Convex Sets and Convex Functions
    8.15 Facts on Quadratic Forms
    8.16 Facts on Simultaneous Diagonalization
    8.17 Facts on Eigenvalues and Singular Values for One Matrix
    8.18 Facts on Eigenvalues and Singular Values for Two or More Matrices
    8.19 Facts on Alternative Partial Orderings
    8.20 Facts on Generalized Inverses
    8.21 Facts on the Kronecker and Schur Products
    8.22 Notes
    Chapter 9. Norms
    9.1 Vector Norms
    9.2 Matrix Norms
    9.3 Compatible Norms
    9.4 Induced Norms
    9.5 Induced Lower Bound
    9.6 Singular Value Inequalities
    9.7 Facts on Vector Norms
    9.8 Facts on Matrix Norms for One Matrix
    9.9 Facts on Matrix Norms for Two or More Matrices
    9.10 Facts on Matrix Norms for Partitioned Matrices
    9.11 Facts on Matrix Norms and Eigenvalues Involving One Matrix
    9.12 Facts on Matrix Norms and Eigenvalues Involving Two or More Matrices
    9.13 Facts on Matrix Norms and Singular Values for One Matrix
    9.14 Facts on Matrix Norms and Singular Values for Two or More Matrices
    9.15 Facts on Least Squares
    9.16 Notes
    Chapter 10. Functions of Matrices and Their Derivatives
    10.1 Open Sets and Closed Sets
    10.2 Limits
    10.3 Continuity
    10.4 Derivatives
    10.5 Functions of a Matrix
    10.6 Matrix Square Root and Matrix Sign Functions
    10.7 Matrix Derivatives
    10.8 Facts Involving One Set
    10.9 Facts Involving Two or More Sets
    10.10 Facts on Matrix Functions
    10.11 Facts on Functions and Derivatives
    10.12 Notes
    Chapter 11. The Matrix Exponential and Stability Theory
    11.1 Definition of the Matrix Exponential
    11.2 Structure of the Matrix Exponential
    11.3 Explicit Expressions
    11.4 Matrix Logarithms
    11.5 The Logarithm Function
    11.6 Lie Groups
    11.7 Lyapunov Stability Theory
    11.8 Linear Stability Theory
    11.9 The Lyapunov Equation
    11.10 Discrete-Time Stability Theory
    11.11 Facts on Matrix Exponential Formulas
    11.12 Facts on the Matrix Sine and Cosine
    11.13 Facts on the Matrix Exponential for One Matrix
    11.14 Facts on the Matrix Exponential for Two or More Matrices
    11.15 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for One Matrix
    11.16 Facts on the Matrix Exponential and Eigenvalues, Singular Values, and Norms for Two or More Matrices
    11.17 Facts on Stable Polynomials
    11.18 Facts on Stable Matrices
    11.19 Facts on Almost Nonnegative Matrices
    11.20 Facts on Discrete-Time-Stable Polynomials
    11.21 Facts on Discrete-Time-Stable Matrices
    11.22 Facts on Lie Groups
    11.23 Facts on Subspace Decomposition
    11.24 Notes
    Chapter 12. Linear Systems and Control Theory
    12.1 State Space and Transfer Function Models
    12.2 Laplace Transform Analysis
    12.3 The Unobservable Subspace and Observability
    12.4 Observable Asymptotic Stability
    12.5 Detectability
    12.6 The Controllable Subspace and Controllability
    12.7 Controllable Asymptotic Stability
    12.8 Stabilizability
    12.9 Realization Theory
    12.10 Zeros
    12.11 H2 System Norm
    12.12 Harmonic Steady-State Response
    12.13 System Interconnections
    12.14 Standard Control Problem
    12.15 Linear-Quadratic Control
    12.16 Solutions of the Riccati Equation
    12.17 The Stabilizing Solution of the Riccati Equation
    12.18 The Maximal Solution of the Riccati Equation
    12.19 Positive-Semidefinite and Positive-Definite Solutions of the Riccati Equation
    12.20 Facts on Stability, Observability, and Controllability
    12.21 Facts on the Lyapunov Equation and Inertia
    12.22 Facts on Realizations and the H2 System Norm
    12.23 Facts on the Riccati Equation
    12.24 Notes
    Bibliography
    Author Index
    Index
    Back Cover
    Matrix Mathematics
    Skew Reflectors r m i t w H e e k S i a n ReflectorsUnitary O Projectors Positive Semide f i n i Nilpotent Positive Definite t e Herm i i a n t N o r m a l e H g n R a e r m itia n v p I n u o G r e rtible
    Matrix Mathematics Theory, Facts, and Formulas Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD
    Copyright c2009 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, 0X20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Bernstein, Dennis S., 1954– Matrix mathematics: theory, facts, and formulas / Dennis S. Bernstein. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-691-13287-7 (hardcover : alk. paper) ISBN 978-0-691-14039-1 (pbk. : alk. paper) 1. Matrices. 2. Linear systems. I. Title. QA188.B475 2008 512.9’434—dc22 2008036257 British Library Cataloging-in-Publication Data is available This book has been composed in Computer Modern and Helvetica. The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ www.press.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
    To the memory of my parents
    . . . vessels, unable to contain the great light flowing into them, shatter and . . . the remains of the broken vessels fall . . . into the lowest world, where break. they remain scattered and hidden — D. W. Menzi and Z. Padeh, The Tree of Life, Chayyim Vital’s Introduction to the Kabbalah of Isaac Luria, Jason Aaronson, Northvale, 1999 Thor . . . placed the horn to his lips . . . He drank with all his might and kept drinking as long as ever he was able; when he paused to look, he could see that the level had sunk a little, . . . for the other end lay out in the ocean itself. — P. A. Munch, Norse Mythology, AMS Press, New York, 1970
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