矩阵数学：理论、事实与公式（第二版）详解

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《矩阵数学：理论、事实与公式》第二版是由Dennis S. Bernstein所著的一本专业书籍，专为深入探讨矩阵理论、性质和运算提供详尽的阐述。该书是针对那些对线性代数和矩阵分析感兴趣的读者，特别是那些在计算机科学、工程、物理学以及数学领域工作的专业人士。作为英文原版书籍，它涵盖了广泛的矩阵概念，如： 1. **范围与Hermitian群**：书中首先介绍了矩阵的基本概念，包括矩阵的定义和它们在向量空间中的作用。Hermitian群（也称共轭共轭群）是一个重要的子集，由Hermitian矩阵（即满足共轭转置等于自身的矩阵）构成，这在量子力学和实数和复数运算中具有显著意义。 2. **可逆性和正规化**：矩阵的可逆性是矩阵论的核心，讨论了如何判断一个矩阵是否可以被逆，以及如何通过正规化（如Jordan标准形）来简化矩阵操作。 3. **规范正交组**：书中会介绍正交矩阵和单位ary矩阵，这些矩阵保持内积不变，对于数据降维、量子计算等领域至关重要。 4. **正定性与半正定性**：这是衡量矩阵对向量的二次型是否总是非负的关键概念。正定矩阵不仅有广泛的应用在数值优化、统计学和控制理论中，而且其特征值性质也有独特的性质。 5. **幂零矩阵和反射器**：幂零矩阵是指所有次方均变为零的矩阵，而反射器和skew-Hermitian矩阵则涉及矩阵的特殊结构，它们在计算机图形学和信号处理中有实际应用。 6. **投影器**：投影矩阵在几何和数据分析中扮演着基础角色，它们表示将向量投影到特定子空间的过程。 7. **nilpotent矩阵**：这种特殊的矩阵在矩阵幂的形式下行为特殊，它们在动力系统和微分方程理论中有所体现。 8. **索引与参考文献**：全书包含丰富的参考文献和索引，确保读者能够进一步探索相关主题并进行深入研究。《矩阵数学：理论、事实与公式》第二版提供了理论背景、实例分析和实用工具，是一本全面且实用的资源，帮助读者巩固和扩展他们在矩阵领域的知识。无论是为了学术研究还是实际问题解决，这本书都是不可或缺的参考资料。

Preface to the S econd Edition

This second edition of Matrix Mathematics represents a major expansion of the

original work. While the total number of pages is increased 46% from 752 to 1100,

the increase is actually greater since this edition is typeset in a smaller font to

facilitate a manageable physical size.

The second edition expands on the ﬁrst edition in several ways. For example,

the new version includes material on graphs (developed within the framework of

relations and partially ordered sets), as well as alternative partial orderings of

matrices, such as rank subtractivity, star, and generalized L¨owner. This edition also

includes additional material on the Kronecker canonical form and matrix pencils;

realizations of ﬁnite groups; zeros of multi-input, multi-output transfer functions;

identities and inequalities for real and complex numbers; bounds on the roots of

polynomials; convex functions; and vector and matrix norms.

The additional material as well as works published subsequent to the ﬁrst edition

increased the number of cited works from 820 to 1503, an increase of 83%. To

increase the utility of the bibliography, this edition uses the “back reference” feature

of LATEX, which indicates where each reference is cited in the text. As in the ﬁrst

edition, the second edition includes an author index. The expansion of the ﬁrst

edition resulted in an increase in the size of the index from 108 pages to 156 pages.

The ﬁrst edition included 57 problems, while the current edition has 73. These

problems represent various extensions or generalizations of known results, some-

times motivated by gaps in the literature.

In this edition, I have attempted to correct all errors that appeared in the ﬁrst

edition. As with the ﬁrst edition, readers are encouraged to contact me about errors

or omissions in the current edition, which I will periodically update on my home

page.

Acknowledgments

I am grateful to many individuals who graciously provided useful advice and ma-

terial for this edition. Some readers alerted me to errors, while others suggested

additional material. In other cases I sought out researchers to help me understand

the precise nature of interesting results. At the risk of omitting those who were help-

ful, I am pleased to acknowledge the following: Mark Balas, Jason Bernstein, Vijay

Chellaboina, Sever Dragomir, Harry Dym, Masatoshi Fujii, Rishi Graham, Was-

sim Haddad, Nicholas Higham, Diederich Hinrichsen, Iman Izadi, Pierre Kabamba,

Preface to the First Edition

The idea for this book began with the realization that at the heart of the solution

to many problems in science, mathematics, and engineering often lies a “matrix

fact,” that is, an identity, inequality, or property of matrices that is crucial to the

solution of the problem. Although there are numerous excellent books on linear

algebra and matrix theory, no one book contains all or even most of the vast number

of matrix facts that appear throughout the scientiﬁc, mathematical, and engineering

literature. This book is an attempt to organize many of these facts into a reference

source for users of matrix theory in diverse applications areas.

Viewed as an extension of scalar mathematics, matrix mathematics provides the

means to manipulate and analyze multidimensional quantities. Matrix mathematics

thus provides powerful tools for a broad range of problems in science and engineer-

ing. For example, the matrix-based analysis of systems of ordinary diﬀerential equa-

tions accounts for interaction among all of the state variables. The discretization of

partial diﬀerential equations by means of ﬁnite diﬀerences and ﬁnite elements yields

linear algebraic or diﬀerential equations whose matrix structure reﬂects the nature

of physical solutions [1238]. Multivariate probability theory and statistical analysis

use matrix methods to represent probability distributions, to compute moments,

and to perform linear regression for data analysis [504, 606, 654, 702, 947, 1181].

The study of linear diﬀerential equations [691, 692, 727] depends heavily on matrix

analysis, while linear systems and control theory are matrix-intensive areas of en-

gineering [3, 65, 142, 146, 311, 313, 348, 371, 373, 444, 502, 616, 743, 852, 865, 935,

1094, 1145, 1153, 1197, 1201, 1212, 1336, 1368, 1455, 1498]. In addition, matrices

are widely used in rigid body dynamics [26, 726, 733, 789, 806, 850, 970, 1026,

1068, 1069, 1185, 1200, 1222, 1351], structural mechanics [863, 990, 1100], compu-

tational ﬂuid dynamics [305, 479, 1426], circuit theory [30], queuing and stochastic

systems [642, 919, 1034], econometrics [403, 948, 1119], geodesy [1241], game theory

[225, 898, 1233], computer graphics [62, 498], computer vision [941], optimization

[255, 374, 953], signal processing [702, 1163, 1361], classical and quantum infor-

mation theory [353, 702, 1042, 1086], communications systems [778, 779], statistics

[580, 654, 948, 1119, 1177], statistical mechanics [16, 159, 160, 1372], demography

[297, 805], combinatorics, networks, and graph theory [165, 128, 179, 223, 235, 266,

269, 302, 303, 335, 363, 405, 428, 481, 501, 557, 602, 702, 844, 920, 931, 1143, 1387],

optics [549, 659, 798], dimensional analysis [641, 1252], and number theory [841].

In all applications involving matrices, computational techniques are essential for

obtaining numerical solutions. The development of eﬃcient and reliable algorithms

for matrix computations is therefore an important area of research that has been

xviii PREFACE TO THE FIRST EDITION

extensively developed [95, 304, 396, 569, 681, 683, 721, 752, 1224, 1225, 1227, 1229,

1315, 1369, 1427, 1431, 1433, 1478]. To facilitate the solution of matrix problems,

entire computer packages have been developed using the language of matrices. How-

ever, this book is concerned with the analytical properties of matrices rather than

their computational aspects.

This book encompasses a broad range of fundamental questions in matrix the-

ory, which, in many cases can be viewed as extensions of related questions in scalar

mathematics. A few such questions follow.

What are the basic properties of matrices? How can matrices be

characterized, classiﬁed, and quantiﬁed?

How can a matrix be decomposed into simpler matrices? A matrix

decomposition may involve addition, multiplication, and partition.

Decomposing a matrix into its fundamental components provides

insight into its algebraic and geometric properties. For example, the

polar decomposition states that every square matrix can be written

as the product of a rotation and a dilation analogous to the polar

representation of a complex number.

Given a pair of matrices having certain properties, what can be

inferred about the sum, product, and concatenation of these matrices?

In particular, if a matrix has a given property, to what extent does that

property change or remain unchanged if the matrix is perturbed by

another matrix of a certain type by means of addition, multiplication,

or concatenation? For example, if a matrix is nonsingular, how large

can an additive perturbation to that matrix be without the sum

becoming singular?

How can properties of a matrix be determined by means of simple

operations? For example, how can the location of the eigenvalues of a

matrix be estimated directly in terms of the entries of the matrix?

To what extent do matrices satisfy the formal properties of the real

numbers? For example, while 0 ≤ a ≤ b implies that a

≤ b

for real

numbers a, b and a positive integer r,whendoes0≤ A ≤ B imply

≤ B

for positive-semideﬁnite matrices A and B andwiththe

positive-semideﬁnite ordering?

Questions of these types have occupied matrix theorists for at least a century,

with motivation from diverse applications. The existing scope and depth of knowl-

edge are enormous. Taken together, this body of knowledge provides a powerful

framework for developing and analyzing models for scientiﬁc and engineering ap-

plications.

This book is intended to be useful to at least four groups of readers. Since

linear algebra is a standard course in the mathematical sciences and engineering,

graduate students in these ﬁelds can use this book to expand the scope of their

PREFACE TO THE FIRST EDITION xix

linear algebra text. For instructors, many of the facts can be used as exercises to

augment standard material in matrix courses. For researchers in the mathematical

sciences, including statistics, physics, and engineering, this book can be used as

a general reference on matrix theory. Finally, for users of matrices in the applied

sciences, this book will provide access to a large body of results in matrix theory.

By collecting these results in a single source, it is my hope that this book will prove

to be convenient and useful for a broad range of applications. The material in this

book is thus intended to complement the large number of classical and modern texts

and reference works on linear algebra and matrix theory [10, 376, 503, 540, 541,

558, 586, 701, 790, 872, 939, 956, 963, 1008, 1045, 1051, 1098, 1143, 1194, 1238].

After a review of mathematical preliminaries in Chapter 1, fundamental proper-

ties of matrices are described in Chapter 2. Chapter 3 summarizes the major classes

of matrices and various matrix transformations. In Chapter 4 we turn to polyno-

mial and rational matrices whose basic properties are essential for understanding

the structure of constant matrices. Chapter 5 is concerned with various decompo-

sitions of matrices including the Jordan, Schur, and singular value decompositions.

Chapter 6 provides a brief treatment of generalized inverses, while Chapter 7 de-

scribes the Kronecker and Schur product operations. Chapter 8 is concerned with

the properties of positive-semideﬁnite matrices. A detailed treatment of vector and

matrix norms is given in Chapter 9, while formulas for matrix derivatives are given

in Chapter 10. Next, Chapter 11 focuses on the matrix exponential and stability

theory, which are central to the study of linear diﬀerential equations. In Chapter

12 we apply matrix theory to the analysis of linear systems, their state space re-

alizations, and their transfer function representation. This chapter also includes a

discussion of the matrix Riccati equation of control theory.

Each chapter provides a core of results with, in many cases, complete proofs.

Sections at the end of each chapter provide a collection of Facts organized to cor-

respond to the order of topics in the chapter. These Facts include corollaries and

special cases of results presented in the chapter, as well as related results that go

beyond the results of the chapter. In some cases the Facts include open problems,

illuminating remarks, and hints regarding proofs. The Facts are intended to provide

the reader with a useful reference collection of matrix results as well as a gateway

to the matrix theory literature.

Acknowledgments

The writing of this book spanned more than a decade and a half, during which

time numerous individuals contributed both directly and indirectly. I am grate-

ful for the helpful comments of many people who contributed technical material

and insightful suggestions, all of which greatly improved the presentation and con-

tent of the book. In addition, numerous individuals generously agreed to read

sections or chapters of the book for clarity and accuracy. I wish to thank Jasim

Ahmed, Suhail Akhtar, David Bayard, Sanjay Bhat, Tony Bloch, Peter Bullen,

Steve Campbell, Agostino Capponi, Ramu Chandra, Jaganath Chandrasekhar,

Nalin Chaturvedi, Vijay Chellaboina, Jie Chen, David Clements, Dan Davison,

Dimitris Dimogianopoulos, Jiu Ding, D. Z. Djokovic, R. Scott Erwin, R. W. Fare-

brother, Danny Georgiev, Joseph Grcar, Wassim Haddad, Yoram Halevi, Jesse

Hoagg, Roger Horn, David Hyland, Iman Izadi, Pierre Kabamba, Vikram Kapila,

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