Scalar, Vector, and Matrix Mathematics

Transcription

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2 Scalar, Vector, and Matrix Mathematics

3 Skew Reflectors Nilpotent O Projectors Skew Hermitian Unitary Reflectors Positive Semidefinite Positive Definite Hermitian Normal Range Hermitian Group Invertible

4 Scalar, Vector, and Matrix Mathematics Theory, Facts, and Formulas Revised and Expanded Edition Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

5 Copyright c 2018 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, 0X20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Bernstein, Dennis S., 1954 Bernstein, Dennis S., 1954 Matrix mathematics. Title: Scalar, vector, and matrix mathematics: theory, facts, and formulas / Dennis S. Bernstein. Other titles: Matrix mathematics Description: Revised and expanded edition. Princeton: Princeton University Press, [2018] Revised and expanded edition of Matrix mathematics, retitled Scalar, vector, and matrix mathematics Preface. Includes bibliographical references and index. Identifiers: LCCN ISBN (hardcover: alk. paper) ISBN (pbk.) Subjects: LCSH: Matrices. Linear systems. Vector analysis. Classification: LCC QA188.B DDC 512.9/434 dc23 LC record available at British Library Cataloging-in-Publication Data is available This book has been composed in Times New Roman 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. press.princeton.edu Printed in the United States of America

6 To the memory of my parents, Irma Shorrie (Hirshon) Bernstein and Milton Bernstein, whose love and guidance are everlasting To Susan, with love and gratitude

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8 ... vessels, unable to contain the great light flowing into them, shatter and break.... the remains of the broken vessels fall... into the lowest world, where 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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10 Contents Preface to the Revised and Expanded Edition Preface to the Second Edition Preface to the First Edition Special Symbols Conventions, Notation, and Terminology xvii xix xxi xxv xxxvii 1. Sets, Logic, Numbers, Relations, Orderings, Graphs, and Functions Sets Logic Relations and Orderings Directed and Symmetric Graphs Numbers Functions and Their Inverses Facts on Logic Facts on Sets Facts on Graphs Facts on Functions Facts on Integers Facts on Finite Sums Facts on Factorials Facts on Finite Products Facts on Numbers Facts on Binomial Coefficients Facts on Fibonacci, Lucas, and Pell Numbers Facts on Arrangement, Derangement, and Catalan Numbers Facts on Cycle, Subset, Eulerian, Bell, and Ordered Bell Numbers Facts on Partition Numbers, the Totient Function, and Divisor Sums Facts on Convex Functions Notes Equalities and Inequalities Facts on Equalities and Inequalities in One Variable Facts on Equalities and Inequalities in Two Variables Facts on Equalities and Inequalities in Three Variables Facts on Equalities and Inequalities in Four Variables 177

11 x CONTENTS 2.5 Facts on Equalities and Inequalities in Five Variables Facts on Equalities and Inequalities in Six Variables Facts on Equalities and Inequalities in Seven Variables Facts on Equalities and Inequalities in Eight Variables Facts on Equalities and Inequalities in Nine Variables Facts on Equalities and Inequalities in Sixteen Variables Facts on Equalities and Inequalities in n Variables Facts on Equalities and Inequalities in 2n Variables Facts on Equalities and Inequalities in 3n Variables Facts on Equalities and Inequalities in 4n Variables Facts on Equalities and Inequalities for the Logarithm Function Facts on Equalities for Trigonometric Functions Facts on Inequalities for Trigonometric Functions Facts on Equalities and Inequalities for Inverse Trigonometric Functions Facts on Equalities and Inequalities for Hyperbolic Functions Facts on Equalities and Inequalities for Inverse Hyperbolic Functions Facts on Equalities and Inequalities in Complex Variables Notes Basic Matrix Properties Vectors Matrices Transpose and Inner Product Geometrically Defined Sets Range and Null Space Rank and Defect Invertibility The Determinant Partitioned Matrices Majorization Facts on One Set Facts on Two or More Sets Facts on Range, Null Space, Rank, and Defect Facts on the Range, Rank, Null Space, and Defect of Partitioned Matrices Facts on the Inner Product, Outer Product, Trace, and Matrix Powers Facts on the Determinant Facts on the Determinant of Partitioned Matrices Facts on Left and Right Inverses Facts on the Adjugate Facts on the Inverse Facts on Bordered Matrices Facts on the Inverse of Partitioned Matrices Facts on Commutators Facts on Complex Matrices Facts on Majorization Notes 362

12 CONTENTS xi 4. Matrix Classes and Transformations Types of Matrices Matrices Related to Graphs Lie Algebras Abstract Groups Addition Groups Multiplication Groups Matrix Transformations Projectors, Idempotent Matrices, and Subspaces Facts on Elementary, Group-Invertible, Range-Hermitian, Range-Disjoint, and Range-Spanning Matrices Facts on Normal, Hermitian, and Skew-Hermitian Matrices Facts on Linear Interpolation Facts on the Cross Product Facts on Inner, Unitary, and Shifted-Unitary Matrices Facts on Rotation Matrices Facts on One Idempotent Matrix Facts on Two or More Idempotent Matrices Facts on One Projector Facts on Two or More Projectors Facts on Reflectors Facts on Involutory Matrices Facts on Tripotent Matrices Facts on Nilpotent Matrices Facts on Hankel and Toeplitz Matrices Facts on Tridiagonal Matrices Facts on Triangular, Hessenberg, and Irreducible Matrices Facts on Matrices Related to Graphs Facts on Dissipative, Contractive, Cauchy, and Centrosymmetric Matrices Facts on Hamiltonian and Symplectic Matrices Facts on Commutators Facts on Partial Orderings Facts on Groups Facts on Quaternions Notes Geometry Facts on Angles, Lines, and Planes Facts on Triangles Facts on Polygons and Polyhedra Facts on Polytopes Facts on Circles, Ellipses, Spheres, and Ellipsoids Polynomial Matrices and Rational Transfer Functions Polynomials Polynomial Matrices The Smith Form and Similarity Invariants Eigenvalues Eigenvectors 511

13 xii CONTENTS 6.6 The Minimal Polynomial Rational Transfer Functions and the Smith-McMillan Form Facts on Polynomials and Rational Functions Facts on the Characteristic and Minimal Polynomials Facts on the Spectrum Facts on Graphs and Nonnegative Matrices Notes Matrix Decompositions Smith Decomposition Reduced Row Echelon Decomposition Multicompanion and Elementary Multicompanion Decompositions Jordan Decomposition Schur Decomposition Singular Value Decomposition, Polar Decomposition, and Full-Rank Factorization Eigenstructure Properties Pencils and the Kronecker Canonical Form Facts on the Inertia Facts on Matrix Transformations for One Matrix 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 Pencils Facts on Eigenstructure for One Matrix Facts on Eigenstructure for Two or More Matrices Facts on Matrix Factorizations Facts on Companion, Vandermonde, Circulant, Permutation, and Hadamard Matrices Facts on Simultaneous Transformations Facts on Additive Decompositions Notes 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 Moore-Penrose Generalized Inverse for Range-Hermitian, Range-Disjoint, and Range-Spanning Matrices Facts on the Moore-Penrose Generalized Inverse for Normal Matrices, Hermitian Matrices, and Partial Isometries Facts on the Moore-Penrose Generalized Inverse for Idempotent Matrices Facts on the Moore-Penrose Generalized Inverse for Projectors Facts on the Moore-Penrose Generalized Inverse for Partitioned Matrices Facts on the Drazin and Group Generalized Inverses for One Matrix Facts on the Drazin and Group Generalized Inverses for Two or More Matrices Facts on the Drazin and Group Generalized Inverses for Partitioned Matrices Notes 679

14 CONTENTS xiii 9. 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 Positive-Semidefinite Matrices Positive-Semidefinite and Positive-Definite Orderings Submatrices and Schur Complements Simultaneous Diagonalization Eigenvalue Inequalities Exponential, Square Root, and Logarithm of Hermitian Matrices Matrix Inequalities Facts on Range and Rank Facts on Unitary Matrices and the Polar Decomposition Facts on Structured Positive-Semidefinite Matrices Facts on Equalities and Inequalities for One Matrix Facts on Equalities and Inequalities for Two or More Matrices Facts on Equalities and Inequalities for Partitioned Matrices Facts on the Trace for One Matrix Facts on the Trace for Two or More Matrices Facts on the Determinant for One Matrix Facts on the Determinant for Two or More Matrices Facts on Convex Sets and Convex Functions Facts on Quadratic Forms for One Matrix Facts on Quadratic Forms for Two or More Matrices Facts on Simultaneous Diagonalization Facts on Eigenvalues and Singular Values for One Matrix Facts on Eigenvalues and Singular Values for Two or More Matrices Facts on Alternative Partial Orderings Facts on Generalized Inverses Facts on the Kronecker and Schur Products Notes Norms Vector Norms Matrix Norms Compatible Norms Induced Norms Induced Lower Bound Singular Value Inequalities Facts on Vector Norms Facts on Vector p-norms Facts on Matrix Norms for One Matrix Facts on Matrix Norms for Two or More Matrices Facts on Matrix Norms for Commutators 884

15 xiv CONTENTS Facts on Matrix Norms for Partitioned Matrices Facts on Matrix Norms and Eigenvalues for One Matrix Facts on Matrix Norms and Eigenvalues for Two or More Matrices Facts on Matrix Norms and Singular Values for One Matrix Facts on Matrix Norms and Singular Values for Two or More Matrices Facts on Linear Equations and Least Squares Notes Functions, Limits, Sequences, Series, Infinite Products, and Derivatives Open Sets and Closed Sets Limits of Sequences Series, Power Series, and Bi-power Series Continuity Derivatives Complex-Valued Functions Infinite Products Functions of a Matrix Matrix Square Root and Matrix Sign Functions Vector and Matrix Derivatives Facts on One Set Facts on Two or More Sets Facts on Functions Facts on Functions of a Complex Variable Facts on Functions of a Matrix Facts on Derivatives Facts on Limits of Functions Facts on Limits of Sequences and Series Notes Infinite Series, Infinite Products, and Special Functions Facts on Series for Subset, Eulerian, Partition, Bell, Ordered Bell, Bernoulli, Euler, and Up/Down Numbers Facts on Bernoulli, Euler, Chebyshev, Legendre, Laguerre, Hermite, Bell, Ordered Bell, Harmonic, Fibonacci, and Lucas Polynomials Facts on the Zeta, Gamma, Digamma, Generalized Harmonic, Dilogarithm, and Dirichlet L Functions Facts on Power Series, Laurent Series, and Partial Fraction Expansions Facts on Series of Rational Functions Facts on Series of Trigonometric and Hyperbolic Functions Facts on Series of Binomial Coefficients Facts on Double-Summation Series Facts on Miscellaneous Series Facts on Infinite Products Notes 1092