When first published in 2005, Matrix Mathematics quickly became the essential reference book for users of matrices in all branches of engineering, science, and applied mathematics. In this fully updated and expanded edition, the author brings together the latest results on matrix theory to make this the most complete, current, and easy-to-use book on matrices.

Each chapter describes relevant background theory followed by specialized results. Hundreds of identities, inequalities, and matrix facts are stated clearly and rigorously with cross references, citations to the literature, and illuminating remarks. Beginning with preliminaries on sets, functions, and relations,Matrix Mathematics covers all of the major topics in matrix theory, including matrix transformations; polynomial matrices; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; vector and matrix norms; the matrix exponential and stability theory; and linear systems and control theory. Also included are a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. This significantly expanded edition of Matrix Mathematics features a wealth of new material on graphs, scalar identities and inequalities, alternative partial orderings, matrix pencils, finite groups, zeros of multivariable transfer functions, roots of polynomials, convex functions, and matrix norms.

Covers hundreds of important and useful results on matrix theory, many never before available in any book

Provides a list of symbols and a summary of conventions for easy use

Includes an extensive collection of scalar identities and inequalities

Features a detailed bibliography and author index with page references

Includes an exhaustive subject index with cross-referencing

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

MAA Reviews

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

Monatshefte fur Mathematik fur Mathematik

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!

SIAM News

When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook.
— Philip J. Davis

IEEE Control Systems Magazine

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

SIAM News

When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook.
— Philip J. Davis

IEEE Control Systems Magazine

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

Image

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

Monatshefte fur Mathematik fur Mathematik

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!

MAA Reviews

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

SIAM Review

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

Monatshefte für Mathematik

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!

SIAM News
- Philip J. Davis

When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook.

IEEE Control Systems Magazine
- Paul Van Dooren

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.

Image
- Helmut Lutkepohl

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.

MAA Reviews
- Henry Ricardo

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.

SIAM Review
- David S. Watkins

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.

From the Publisher

"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 für 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

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

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## More About This Textbook

## Overview

When first published in 2005,

Matrix Mathematicsquickly became the essential reference book for users of matrices in all branches of engineering, science, and applied mathematics. In this fully updated and expanded edition, the author brings together the latest results on matrix theory to make this the most complete, current, and easy-to-use book on matrices.Each chapter describes relevant background theory followed by specialized results. Hundreds of identities, inequalities, and matrix facts are stated clearly and rigorously with cross references, citations to the literature, and illuminating remarks. Beginning with preliminaries on sets, functions, and relations,

Matrix Mathematicscovers all of the major topics in matrix theory, including matrix transformations; polynomial matrices; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; vector and matrix norms; the matrix exponential and stability theory; and linear systems and control theory. Also included are a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. This significantly expanded edition ofMatrix Mathematicsfeatures a wealth of new material on graphs, scalar identities and inequalities, alternative partial orderings, matrix pencils, finite groups, zeros of multivariable transfer functions, roots of polynomials, convex functions, and matrix norms.## Editorial Reviews

## Image

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

## MAA Reviews

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

## Monatshefte fur Mathematik fur Mathematik

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!## SIAM News

When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook.— Philip J. Davis

## IEEE Control Systems Magazine

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

## SIAM News

When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook.— Philip J. Davis

## IEEE Control Systems Magazine

Matrix Mathematicscontains 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

## Image

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

## Monatshefte fur Mathematik fur Mathematik

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!## MAA Reviews

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

## SIAM Review

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

## Monatshefte für Mathematik

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!

## SIAM News - Philip J. Davis

When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook.## IEEE Control Systems Magazine - Paul Van Dooren

Matrix Mathematicscontains 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.## Image - Helmut Lutkepohl

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.## MAA Reviews - Henry Ricardo

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.## SIAM Review - David S. Watkins

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.## From the Publisher

"When a matrix question is thrown my way, I will now refer my correspondents . . . to Bernstein's handbook."—Philip J. Davis,SIAM News"

Matrix Mathematicscontains 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 für 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## Product Details

## Related Subjects

## Meet the Author

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

## Table of Contents

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