矩阵分析：卷1（英文影印版）

    书籍
    数学书籍
本文从数学分析的角度阐述了矩阵分析的经典和现代方法，不仅包括由于数学分析的需要而产生的线性代数的论题，还广泛选择了其他相关学科如微分方程、最优化、逼近理论、工程学和运筹学等有关的论题。本书主要内容有：特征值、特征向量和相似性、酉相似、Schur三角化及其推论、正规矩阵、标准形和包括Jordan标准形在内的各种分解、LU分解、QR分解和酉矩阵、Hermite矩阵和复对称矩阵、向量范数和矩阵范数、特征值的估计和扰动、正定矩阵、非负矩阵。
本书由美国著名数学家R.A.Horn教授和C．R．Johnson教授合著，是矩阵理论方面的经典著作，原书自1985年出版以来，已经重印了10余次。书中论述了矩阵分析的经典方法和现代方法，不仅涵盖了几乎所有的基础理论，还广泛地对涉及其他相关学科的各种论题进行了有效的阐述，并对有关论题提供了现代的参考资料。
本书逻辑清晰，结构严谨，既注重教学又注重应用。在每一章的开始，作者都介绍几个应用来引入本章的论题以激发学习兴趣。在章节末尾，作者还独具匠心地编排了许多具有探索性和启发性的习题，引导读者提高描述和解决数学问题的能力。所以不论是对从事线性代数纯理论研究还是从事应用研究的人员，本书都是一本必备的参考书。
本书可作为理工科专业研究生或数学专业高年级本科生教材，也可供数学工作者和科技人员参考。

Rogeer A，horn 线性代数和矩阵理论领域国际知名权威，1967年获得斯坦福大学数学博士，197-1979年任约翰·霍普金斯大学数学系系主任，现为犹他大学教授。曾担任American Mathematical Monthly编辑.
Charles R．Johnson线性代数和矩阵理论领域国际知名权威。现为威廉玛丽大学教授。曾发表近300篇论文因其在数学科学领域的杰出贡献被授予华盛顿科学学会奖，担任过所有主要矩外分析类杂忐的编和两份SIAM杂志的主编。

Contents
Chapter 0 Review and miscellanea　1
0.0　Introduction　1
0.1　Vector spaces　1
0.2　Matrices　4
0.3　Determinants　7
0.4　Rank　12
0.5　Nonsingularity　14
0.6　The usual inner product　14
0.7　Partitioned matrices　17
0.8　Determinants again　19
0.9　Special types of matrices　23
0.10 Change of basis 30
Chapter 1 Eigenvalues, eigenvectors, and similarity　33
1.0　Introduction　33
1.1　The eigenvalue-eigenvector equation　34
1.2　The characteristic polynomial　38
1.3　Similarity　44
1.4　Eigenvectors　57
Chapter 2 Unitary equivalence and normal matrices　65

　　Linear algebra and matrix theory have long been fundamental tools inmathematical disciplines as well as fertile fields for research in their ownright. In this book, and in the companion volume, Topics in MatrixAna-iysis, we present classical and recent results of matrix analysis that haveproved to be important to applied mathematics. The book may be usedas an undergraduate or graduate text and as a self-contained referencefor a variety of audiences. We assume background equivalent to a one-semester elementary linear algebra course and knowledge of rudimentaryanalytical concepts. We begin with the notions of eigenvalues and eigen-vectors; no prior knowledge of these concepts is assumed.
　　Facts about matrices, beyond those found in an elementary linear alge-bra course, are necessary to understand virtually any area of mathemati-cal science, whether it be differential equations; probability and statistics;optimization; or applications in theoretical and applied economics, theengineering disciplines, or operations research, to name only a few. Butuntil recently, much of the necessary material has occurred sporadically(or not at all) in the undergraduate and graduate curricula. As interest inapplied mathematics has grown and more courses have been devoted toadvanced matrix theory, the need for a text offering a broad selection oftopics has become more apparent, as has the need for a modern referenceon the subject.
　　There are a number of well-loved classics in matrix theory, but theyare not well suited for general classroom use, nor for systematic individ-ual study. A lack of problems, applications, and motivation; an inade-quate index; and a dated approach are among the difficulties confrontingreaders of some traditional references. More recent books tend to be eitherelementary texts or treatises devoted to special topics. Our goal was towrite a book that would be a useful modern treatment of a broad rangeof topics.
　　One view of "matrix analysis" is that it consists of those topics inlinear algebra that have arisen out of the needs of mathematical analysis,such as multivariable calculus, complex variables, differential equations,optimization, and approximation theory. Another view is that matrixanalysis is an approach to real and complex linear algebraic problemsthat does not hesitate to use notions from analysis such as limits, con-tinuity, and power series when these seem more efficient or natural thana purely algebraic approach. Both views of matrix analysis are reflectedin the choice and treatment of topics in this book. We prefer the termmatrix analysis to linear algebra as an accurate reflection of the broadscope and methodology of the field.
　　For review and convenience in reference, Chapter 0 contains a sum-mary of necessary facts from elementary linear algebra, as well as otheruseful, though not necessarily elementary, facts. Chapters 1, 2, and 3contain mainly core material likely to be included in any second course inlinear algebra or matrix theory: a basic treatment of eigenvalues, eigen-vectors, and similarity; unitary similarity, Schur triangularization and itsimplications, and normal matrices; and canonical forms and factoriza-tions including the Jordan form, LU factorization, OR factorization,and companion matrices. Beyond this, each chapter is developed sub-stantially independently and treats in some depth a major topic:
　　Hermitian and complex symmetric matrices (Chapter 4). We give special emphasis to variational methods for studying eigenvalues of Hermitian matrices and include an introduction to the notion of majorization.
　　Norms on vectors and matrices (Chapter 5) are essential for er ror analyses of numerical linear algebraic algorithms and for the study of matrix power series and iterative processes. We discuss the algebraic, geometric, and analytic properties of norms in some detail, and make a careful distinction between those norm results for matrices that depend on the submultiplicativity axiom for matrix norms and those that do not.
　　Eigenvalue location and perturbation results (Chapter 6) for gen eral (not necessarily Hermitian) matrices are important for many applications. We give a detailed treatment of the theory of Gers gorin regions, and some of its modern refinements, and of rele vant graph theoretic concepts.
　　Positive definite matrices (Chapter 7) and their applications, in cluding inequalities, are considered at some length. A discussion of the polar and singular value decompositions is included, along with applications to matrix approximation problems.
　　Component-wise nonnegative and positive matrices (Chapter 8) arise in many applications in which nonnegative quantities nec essarily occur (probability, economics, engineering, etc.), and their remarkable theory reflects the applications. Our develop ment of the theory of nonnegative, positive, primitive, and irre ducible matrices proceeds in elementary steps based upon the use of norms.
　　In the companion volume, further topics of similar interest are treated:the field of values and generalizations; inertia, stable matrices, M-matricesand related special classes; matrix equations, Kronecker and Hadamardproducts; and various ways in which functions and matrices may belinked.
　　This book provides the basis for a variety of oneor two-semestercourses through selection of chapters and sections appropriate to a par-ticular audience. We recommend that an instructor make a careful pre-selection of sections and portions of sections of the book for the needs ofa particular course. This would probably include Chapter 1, much ofChapters 2 and 3, and facts about Hermitian matrices and norms fromChapters 4 and 5.
　　Most chapters.contain some relatively specialized or nontraditionalmaterial. For example, Chapter 2 includes not only Schur's basic theoremon unitary triangularization of a single matrix, but also a discussion ofsimultaneous triangularization of families of matrices. In the section onunitary equivalence, our presentation of the usual facts is followed by adiscussion of trace conditions for two matrices to be unitarily equivalent.A discussion of complex symmetric matrices in Chapter 4 provides acounterpoint to the development of the classical theory of Hermitianmatrices. Basic aspects of a topic appear in the initial sections of eachchapter, while more elaborate discussions occur at the ends of sections orin later sections. This strategy has the advantage of presenting topics in asequence that enhances the book's utility as a reference. It also providesa rich variety of options to the instructor.
　　Many of the results discussed hold or can be generalized to hold formatrices over other fields or in some broader algebraic setting. However,we deliberately confine our domain to the real and complex fields wherefamiliar methods of classical analysis as well as formal algebraic tech-niques may be employed.
　　Though we generally consider matrices to have complex entries, mostexamples are confined to real matrices, and no deep knowledge of com-plex analysis is required. Acquaintance with the arithmetic of complexnumbers is necessary for an understanding of matrix analysis and iscovered to the extent necessary in an appendix. Other brief appendicescover several peripheral, but essential, topics such as Weierstrass's theo-rem and convexity.
　　We have included many exercises and problems because we feel theseare essential to the development of an understanding of the subject andits implications. The exercises occur throughout as part of the develop-ment of each section; they are generally elementary and of immediate usein understanding the concepts. We recommend that the reader work atleast a broad selection of these. Problems are listed (in no particularorder) at the end of each section; they cover a range of difficulties andtypes (from theoretical to computational) and they may extend the topic,develop special aspects, or suggest alternate proofs of major ideas. Sig-nificant hints are given for the more difficult problems. The results ofsome problems are referred to in other problems or in the text itself. Wecannot overemphasize the importance of the reader's active involvementin carrying out the exercises and solving problems.
　　While the book itself is not about applications, we have, for motiva-tional purposes, begun each chapter with a section outlining a few appli-cations to introduce the topic of the chapter.
　　Readers who wish to consult alternate treatments of a topic for ad-ditional information Ere referred to the books listed in the Referencessection following the appendices. These books are cited in the text usinga brief mnemonic code; for example, a book by Jones and Smith mightbe referred to as [JSm]. The codes and complete citations appear alpha-betically by author in the References section.
　　The list of book references is not exhaustive. As a practical concessionto the limits of space in a general multitopic book, we have minimizedthe number of citations in the text. A small selection of references to papers such as those we have explicitly used does occur at the end of most sections accompanied by a brief discussion, but we have made no attempt to collect historical references to classical results. Extensive bibliographies are provided in the more specialized books we have referenced. The reader should also be aware of broad and current bibliographical resources covering portions of matrix analysis such as the KWIC Index for Numerical Linear Algebra [CaLe] and sections 15 and 65 of the Mathematical Reviews.
　　We appreciate the helpful suggestions of our colleagues and students who have taken the time to convey their reactions to the class notes andpreliminary manuscripts that were the precursors of the book. They in-clude Wayne Barrett, Leroy Beasley, Bryan Cain, David Carlson, DipaChoudhury, Risana Chowdhury, Yoo Pyo Hong, Dmitry Krass, DaleOlesky, Stephen Pierce, Leiba Rodman, and Pauline van den Driessche.