陶有田078200333-英文及翻译
1 view
Skip to first unread message
风继续吹
Dec 3, 2007, 9:54:11 AM12/3/07
to drenglish
Preface
Linear algebra and matrix theory have long been fundamental tools
in mathematical disciplines as well as fertile fields for research in
their own right. In this book, and in the companion volume, Topics in
Matrix Analysis, we present classical and recent results of matrix
analysis that have proved to be important to applied mathematics. The
book may be used as an undergraduate or graduate text and as a self-
contained reference for a variety of audiences. We assume background
equivalent to a one-semester elementary linear algebra course and
knowledge of rudimentary analytical concepts. We begin with the
notions of eigenvalues and eigenvectors; no prior knowledge of these
concepts is assumed.
Facts about matrices, beyond those found in an elementary linear
algebra course, are necessary to understand virtually any area of
mathematical science, whether it be differential equations;
probability and statistics; optimization; or applications in
theoretical and applied economics, the engineering disciplines, or
operations research, to name only a few. But until recently, much of
the necessary material has occurred sporadically (or not at all) in
the undergraduate and graduate curricula. As interest in applied
mathematics has grown and more courses have been devoted to advanced
matrix theory, the need for a text offering a broad selection of
topics has become more apparent, as has the need for a modern
reference on the subject.
There are a number of well-loved classics in matrix theory, but
they are not well suited for general classroom use, nor for systematic
individual study. A lack of problems, applications, and motivation; an
inadequate index; and a dated approach are among the difficulties
confronting readers of some traditional references. More recent books
tend to be either elementary texts or treatises devoted to special
topics. Our goal was to write a book that would be a useful modern
treatment of a broad range of topics.
One view of "matrix analysis" is that it consists of those topics
in linear 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 matrix analysis is an approach to real and
complex linear algebraic problems that does not hesitate to use
notions from analysis-such as limits, continuity, and power series-
when these seem more efficient or natural than a purely algebraic
approach. Both views of matrix analysis are reflected in the choice
and treatment of topics in this book. We prefer the term matrix
analysis to linear algebra as an accurate reflection of the broad
scope and methodology of the field.
For review and convenience in reference, Chapter 0 contains a
summary of necessary facts from elementary linear algebra, as well as
other useful, though not necessarily elementary, facts. Chapter1,2,
and 3 contain mainly core material likely to be included in any second
course in linear algebra or matrix theory: a basic treatment of
eigenvalues, eigenvectors, and similarity; unitary similarity, Schur
triangularization and its implications, and normal matrices; and
canonical forms and factorizations including the Jordan form, LU
factorization, QR factorization, and companion matrices. Beyond this,
each chapter is developed substantially 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 error
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
general (not necessarily Hermitian) matrices are important for many
applications. We give a detailed treatment of the theory of GerŠgorin
regions, and some of its modern refinements, and of relevant graph
theoretic concepts.
Positive definite matrices (Chapter 7) and their applications,
including 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 necessarily occur
(probability, economics, engineering, etc.) and their remarkable
theory reflects the applications. Our development of the theory of
nonnegative, positive, and irreducible matrices proceeds on 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-matrices and related special classes; matrix equations,
Kronecker and Hadamard products; and various ways in which functions
and matrices may be linked.
This book provides the basis for a variety of one- or two-semester
courses through selection of chapters and sections appropriate to a
particular audience. We recommend that an instructor make a careful
preselection of sections and portions of sections of the book for the
needs of a particular course. This would probably include Chapter 1,
much of Chapters 2 and 3, and facts about Hermitian matrices and norms
from Chapters 4 and 5.
Most chapters contain some relatively specialized or nontraditional
material. For example, Chapter 2 includes not only Schur's basic
theorem on unitary triangularization of a single matrix, but also a
discussion of simultaneous triangularization of families of matrices.
In the section on unitary equivalence, our presentation of the usual
facts is followed y a discussion of trace conditions for two matrices
to be unitarily equivalent. A discussion of complex symmetric matrices
in Chapter 4 provides a counterpoint to the development of the
classical theory of Hermitian matrices. Basic aspects of a topic
appear in the initial sections of each chapter, while more elaborate
discussions occur at the ends of sections or in later sections. This
strategy has the advantage of presenting topics in a sequence that
enhances the book's utility as a reference. It also provides a rich
variety of options to the instructor.
Many of the results discussed hold or can be generalized to hold for
matrices over other fields or in some broader algebraic setting.
However, we deliberately confine our domain to the real and complex
fields where familiar methods of classical analysis as well as formal
algebraic techniques may e employed.
Though we generally consider matrices to have complex entries, most
examples are confined to real matrices, and no deep knowledge of
complex analysis is required. Acquaintance with the arithmetic of
complex numbers is necessary for an understanding of matrix
analysis and is covered to the extent necessary in an appendix. Other
brief appendices cover several peripheral, but essential, topics such
as Weierstrass's theorem and convexity.
We have included many exercises and problems because we feel these
are essential to the development of an understanding of the subject
and its implications. The exercises occur throughout as part of the
development of each section; they are generally elementary and of
immediate use in understanding the concepts. We recommend that the
reader work at least a broad selection of these. Problems are listed
(in no particular order) at the end of each section; they cover a
range of difficulties and types (from theoretical to computational)
and they may extend the topic, develop special aspects, or suggest
alternate proofs of major ideas. Significant hints are given for the
more difficult problems. The results of some problems are referred to
in other problems or in the text itself. We cannot overemphasize the
importance of the reader's active involvement in carrying out the
exercises and solving problems.
While the book itself is not about applications, we have, for
motivational purposes, begun each chapter with a section outlining a
few applications to introduce the topic of the chapter.
Readers who wish to consult alternate treatments of a topic for
additional information are referred to the books listed in the
References section following the appendices. These books are cited in
the text using a brief mnemonic code; for example, a book by Jones and
Smith might be referred to as [JSm]. The codes and complete citations
appear alphabetically by author in the References section.
The list of book references is not exhaustive. As a practical
concession to the limits of space in a general multitopic book, we
have minimized the 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 and preliminary manuscripts that were the precursors of
the book. They include Wayne Barrett, Leroy Beasley, Bryan Cain, David
Carlson, Dipa Choudhury, Risana Chowdhury, Yoo Pyo Hong, Dmitry Krass,
Dale Olesky, Stephen Pierce, Leiba Rodman, and Pauline van den
Driessche.
Roger A. Horn
Charles R. Johnson
译文：前 言
长期以来，在数学的各分支学科及其他多个领域的研究中，线性代数和矩阵理论一直都是有着自身特色的基本的工具。对于应用数学来说，矩阵分析的重
要性已经得到了证明，本书及下一卷《矩阵分析漫谈》介绍了它的经典的和近来的研究成果。本书可作为本科或研究生教材或者自学参考书。我们假设读者已学了
一学期的初等线性代数课程及基本的分析的知识。本书由特征值及特征向量开始，不需要有关这些概念的预备知识。
除了基础线性代数课程中关于矩阵的知识，矩阵理论对于深入理解数学的各领域中是不无帮助的，不论它们是微分方程，概率论与数理统计，最优化理论
还是在理论和应用经济学以及工程学或者运筹学中的应用，此处仅举这几例。但是迄今为止，在本科生及研究生课程里，许多必需的材料零星的出现或根本就没有
出现。由于对于应用数学的兴趣业已增长，更多的课程都在关注高等矩阵理论，因而就显然需要一本能够提供广泛主题的教材，正如需要一本现代的关于矩阵理论
的参考书一样。
关于矩阵理论，已经有许多深受喜爱的经典著作了，但是它们既不适合作为一般的教材，也不适合作为自学参考书。传统参考书的读者所面临的困难在
于，问题、应用及动机的缺乏，不充分的索引和过时的方法。近来更多的书籍趋向于基础教材或者是有着特殊主题的论文。我们的目的就是写一本涉及各种主题的
有用的现代的教材。对于矩阵分析的一种认识，就是它是由线性代数中的各个主题构成的，这些主题是出于对数学分析的需要而出现的，例如多元微积分学，复变
函数，微分方程，最优化理论以及逼近理论。另外一种观点就是，矩阵分析是一种解决实(复)线性代数问题的方法，它会毫不犹豫地使用分析的概念(如极限、
连续性和幂级数)，这种情况会在这些概念比纯代数方法更加有效、简单时发生。上述两种观点在选择及处理本书的主题时均得到了反映。与线性代数相比，我们
更喜欢将矩阵分析作为广泛范围及域方法的一个精确反映。为了复习及出于参考的便利，第0章概括了初等线性代数的基本知识和一些其他有用(但未必是初等
的)的知识。第1章、第2章及第3章主要包括以下核心内容：特征值、特征向量及相似性的基本知识；酉相似，Schur三对角化及其应用，正规矩阵；
Jordan标准形及Jordan分解，LU分解，QR分解及伴随矩阵。此外，各章充分独立，且都深入地研究了某个主要的论题：
Hermite及复矩阵(第4章) 我们特别强调了用变分方法去研究Hermite矩阵，介绍了最优化的概念。
向量及矩阵范数(第5章)是数值线性代数算法中误差分析及幂级数和迭代过程研究的核心内容。我们详细地讨论了范数的代数性质、几何性质及解析
性质，根据下面的要求对矩阵范数结论加以仔细的区别：看这些矩阵是否依据依据矩阵范数次可乘性公理。
在许多应用方面，一般的矩阵(未必为Hermite型的)特征值的估计和扰动结果(第6章)是比较重要的。我们给出了GerŠgorin区域
理论的详细论述及它的一些现代的细化和相关的图表理论概念。
详细考察了正定矩阵及包括不等式在内的应用(第7章)。讨论了极点和奇异值分解以及矩阵逼近问题的应用。
分量非负矩阵及正矩阵(第8章)在许多应用中都可见到，在这些应用中必然会出现非负的量，如概率论，经济学，工程学等，其中的著名理论反映了这
些应用。我们在按照基本的步骤进行以范数应用为基础的非负矩阵、正矩阵、不可约矩阵的理论研究。
在下一卷，我们讨论了进一步的话题：值域及其推广；惯性矩阵，稳定矩阵，M-矩阵及相关的特殊分类；矩阵方程，Kronecker
积，Hadamard积；各种用于函数及矩阵的方法。
通过选择适合于特殊读者的章节，本书为一或两个学期的课程提供了基础。我们建议在教授本书时，教师可有选择的对本书各节内容进行讲授，其中包括第我章，
第2 章、第3章的大部分，第4章、第5章的Hermite矩阵及Hermite范数。
本书大部分章节都包含一些相对专业的、非传统的内容。例如，第2 章不仅包含关于单一矩阵酉三对角化的Schur基本定理，也讨论了一族矩阵的
同步三对角化。基本知识点通常安排在每章的前半部分，而更多的细节安排在每章的后半部分。这种安排方式有利于按照先后顺序介绍知识，并且能提高本书作为
参考书的作用。这样做同样也为授课都是提供了更多的选择。
对于其他域或更广泛代数集上的矩阵而言，许多讨论的结论都成立或经过推广后能够成立。然而，我们有意地将讨论的范围限定于实数和复数域，其中我们使用了
和形式代数方法一样的经典的分析方法。
虽然我们认为矩阵是复矩阵，但大部分的例子都只限定在实数范围内，没有更深入地去讨论复数矩阵的情形。熟悉复数算术对于理解矩阵分析是必要的，
这方面的知识安排在附录中。其他具有概括性的附录包含了几个外围的但非核心的话题，如Weierstrass定理及凸性。
我们给出了许多习题及问题，因为我们觉得对于理解矩阵分析及其应用来说，它们是非常重要的。练习题在每个小节中给出，一般都比较简单，是本小节
概念的直接应用。我们建议读者最好选择其中之一来做。问题通常在每节的结尾(不按特别的次序)列出，它们涵盖了本节的难点和(由理论到计算的)类型，展
开了主题，推广了特殊的知识点，或者暗示了主要思想的不同证明。有一定难度的问题我们都给了重要的暗示。一些问题的结论在其他问题或在教材本身中都有涉
及。我们觉得读者练习及解答问题是比较重要的。
本书附录后列出的参考文献为那些希望看到矩阵分析不同论述的读者提供了额外的信息。本书用易简短的易记符号来引用这些参考书，如Jones和
Smith写的一本书被记为[JSm]。作者将这些参考文献按照字母顺序来进行排列。
参考文献并不详尽。在这样一个多主题书籍中，为了节省有限的篇幅，我们将教材中的引文数量压缩到了最小。我们将诸如那些我们已经明确使用的论文
安排在了大部分章节后的简短讨论中，但是我们没有去搜集历史上的关于经典结论的参考文献。大量的参考书目由我们所参考的更专业的书籍给出了。读者也应该
知道广泛的当前的参考资料，这些资料涵盖了矩阵分析的各个方面，如《数值线性代数的国际标准题内关键词索引》[CaLe]及《数学评论》的第15
节、65节。
对于我们同事及学生的帮助性的建议，我们表示感激。这些学生花了大量的时间去整理课堂笔记及原始手稿，这些都构成了本书的原稿。他们是：Wayne
Barrett，Leroy Beasley，Bryan Cain， David Carlson， Dipa Choudhury，
Risana Chowdhury，Yoo Pyo Hong，Dmitry Krass，Dale Olesky， Stephen
Pierce，Leiba Rodman和Pauline van den Driessche。
Linear algebra and matrix theory have long been fundamental tools
in mathematical disciplines as well as fertile fields for research in
their own right. In this book, and in the companion volume, Topics in
Matrix Analysis, we present classical and recent results of matrix
analysis that have proved to be important to applied mathematics. The
book may be used as an undergraduate or graduate text and as a self-
contained reference for a variety of audiences. We assume background
equivalent to a one-semester elementary linear algebra course and
knowledge of rudimentary analytical concepts. We begin with the
notions of eigenvalues and eigenvectors; no prior knowledge of these
concepts is assumed.
Facts about matrices, beyond those found in an elementary linear
algebra course, are necessary to understand virtually any area of
mathematical science, whether it be differential equations;
probability and statistics; optimization; or applications in
theoretical and applied economics, the engineering disciplines, or
operations research, to name only a few. But until recently, much of
the necessary material has occurred sporadically (or not at all) in
the undergraduate and graduate curricula. As interest in applied
mathematics has grown and more courses have been devoted to advanced
matrix theory, the need for a text offering a broad selection of
topics has become more apparent, as has the need for a modern
reference on the subject.
There are a number of well-loved classics in matrix theory, but
they are not well suited for general classroom use, nor for systematic
individual study. A lack of problems, applications, and motivation; an
inadequate index; and a dated approach are among the difficulties
confronting readers of some traditional references. More recent books
tend to be either elementary texts or treatises devoted to special
topics. Our goal was to write a book that would be a useful modern
treatment of a broad range of topics.
One view of "matrix analysis" is that it consists of those topics
in linear 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 matrix analysis is an approach to real and
complex linear algebraic problems that does not hesitate to use
notions from analysis-such as limits, continuity, and power series-
when these seem more efficient or natural than a purely algebraic
approach. Both views of matrix analysis are reflected in the choice
and treatment of topics in this book. We prefer the term matrix
analysis to linear algebra as an accurate reflection of the broad
scope and methodology of the field.
For review and convenience in reference, Chapter 0 contains a
summary of necessary facts from elementary linear algebra, as well as
other useful, though not necessarily elementary, facts. Chapter1,2,
and 3 contain mainly core material likely to be included in any second
course in linear algebra or matrix theory: a basic treatment of
eigenvalues, eigenvectors, and similarity; unitary similarity, Schur
triangularization and its implications, and normal matrices; and
canonical forms and factorizations including the Jordan form, LU
factorization, QR factorization, and companion matrices. Beyond this,
each chapter is developed substantially 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 error
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
general (not necessarily Hermitian) matrices are important for many
applications. We give a detailed treatment of the theory of GerŠgorin
regions, and some of its modern refinements, and of relevant graph
theoretic concepts.
Positive definite matrices (Chapter 7) and their applications,
including 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 necessarily occur
(probability, economics, engineering, etc.) and their remarkable
theory reflects the applications. Our development of the theory of
nonnegative, positive, and irreducible matrices proceeds on 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-matrices and related special classes; matrix equations,
Kronecker and Hadamard products; and various ways in which functions
and matrices may be linked.
This book provides the basis for a variety of one- or two-semester
courses through selection of chapters and sections appropriate to a
particular audience. We recommend that an instructor make a careful
preselection of sections and portions of sections of the book for the
needs of a particular course. This would probably include Chapter 1,
much of Chapters 2 and 3, and facts about Hermitian matrices and norms
from Chapters 4 and 5.
Most chapters contain some relatively specialized or nontraditional
material. For example, Chapter 2 includes not only Schur's basic
theorem on unitary triangularization of a single matrix, but also a
discussion of simultaneous triangularization of families of matrices.
In the section on unitary equivalence, our presentation of the usual
facts is followed y a discussion of trace conditions for two matrices
to be unitarily equivalent. A discussion of complex symmetric matrices
in Chapter 4 provides a counterpoint to the development of the
classical theory of Hermitian matrices. Basic aspects of a topic
appear in the initial sections of each chapter, while more elaborate
discussions occur at the ends of sections or in later sections. This
strategy has the advantage of presenting topics in a sequence that
enhances the book's utility as a reference. It also provides a rich
variety of options to the instructor.
Many of the results discussed hold or can be generalized to hold for
matrices over other fields or in some broader algebraic setting.
However, we deliberately confine our domain to the real and complex
fields where familiar methods of classical analysis as well as formal
algebraic techniques may e employed.
Though we generally consider matrices to have complex entries, most
examples are confined to real matrices, and no deep knowledge of
complex analysis is required. Acquaintance with the arithmetic of
complex numbers is necessary for an understanding of matrix
analysis and is covered to the extent necessary in an appendix. Other
brief appendices cover several peripheral, but essential, topics such
as Weierstrass's theorem and convexity.
We have included many exercises and problems because we feel these
are essential to the development of an understanding of the subject
and its implications. The exercises occur throughout as part of the
development of each section; they are generally elementary and of
immediate use in understanding the concepts. We recommend that the
reader work at least a broad selection of these. Problems are listed
(in no particular order) at the end of each section; they cover a
range of difficulties and types (from theoretical to computational)
and they may extend the topic, develop special aspects, or suggest
alternate proofs of major ideas. Significant hints are given for the
more difficult problems. The results of some problems are referred to
in other problems or in the text itself. We cannot overemphasize the
importance of the reader's active involvement in carrying out the
exercises and solving problems.
While the book itself is not about applications, we have, for
motivational purposes, begun each chapter with a section outlining a
few applications to introduce the topic of the chapter.
Readers who wish to consult alternate treatments of a topic for
additional information are referred to the books listed in the
References section following the appendices. These books are cited in
the text using a brief mnemonic code; for example, a book by Jones and
Smith might be referred to as [JSm]. The codes and complete citations
appear alphabetically by author in the References section.
The list of book references is not exhaustive. As a practical
concession to the limits of space in a general multitopic book, we
have minimized the 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 and preliminary manuscripts that were the precursors of
the book. They include Wayne Barrett, Leroy Beasley, Bryan Cain, David
Carlson, Dipa Choudhury, Risana Chowdhury, Yoo Pyo Hong, Dmitry Krass,
Dale Olesky, Stephen Pierce, Leiba Rodman, and Pauline van den
Driessche.
Roger A. Horn
Charles R. Johnson
译文：前 言
长期以来，在数学的各分支学科及其他多个领域的研究中，线性代数和矩阵理论一直都是有着自身特色的基本的工具。对于应用数学来说，矩阵分析的重
要性已经得到了证明，本书及下一卷《矩阵分析漫谈》介绍了它的经典的和近来的研究成果。本书可作为本科或研究生教材或者自学参考书。我们假设读者已学了
一学期的初等线性代数课程及基本的分析的知识。本书由特征值及特征向量开始，不需要有关这些概念的预备知识。
除了基础线性代数课程中关于矩阵的知识，矩阵理论对于深入理解数学的各领域中是不无帮助的，不论它们是微分方程，概率论与数理统计，最优化理论
还是在理论和应用经济学以及工程学或者运筹学中的应用，此处仅举这几例。但是迄今为止，在本科生及研究生课程里，许多必需的材料零星的出现或根本就没有
出现。由于对于应用数学的兴趣业已增长，更多的课程都在关注高等矩阵理论，因而就显然需要一本能够提供广泛主题的教材，正如需要一本现代的关于矩阵理论
的参考书一样。
关于矩阵理论，已经有许多深受喜爱的经典著作了，但是它们既不适合作为一般的教材，也不适合作为自学参考书。传统参考书的读者所面临的困难在
于，问题、应用及动机的缺乏，不充分的索引和过时的方法。近来更多的书籍趋向于基础教材或者是有着特殊主题的论文。我们的目的就是写一本涉及各种主题的
有用的现代的教材。对于矩阵分析的一种认识，就是它是由线性代数中的各个主题构成的，这些主题是出于对数学分析的需要而出现的，例如多元微积分学，复变
函数，微分方程，最优化理论以及逼近理论。另外一种观点就是，矩阵分析是一种解决实(复)线性代数问题的方法，它会毫不犹豫地使用分析的概念(如极限、
连续性和幂级数)，这种情况会在这些概念比纯代数方法更加有效、简单时发生。上述两种观点在选择及处理本书的主题时均得到了反映。与线性代数相比，我们
更喜欢将矩阵分析作为广泛范围及域方法的一个精确反映。为了复习及出于参考的便利，第0章概括了初等线性代数的基本知识和一些其他有用(但未必是初等
的)的知识。第1章、第2章及第3章主要包括以下核心内容：特征值、特征向量及相似性的基本知识；酉相似，Schur三对角化及其应用，正规矩阵；
Jordan标准形及Jordan分解，LU分解，QR分解及伴随矩阵。此外，各章充分独立，且都深入地研究了某个主要的论题：
Hermite及复矩阵(第4章) 我们特别强调了用变分方法去研究Hermite矩阵，介绍了最优化的概念。
向量及矩阵范数(第5章)是数值线性代数算法中误差分析及幂级数和迭代过程研究的核心内容。我们详细地讨论了范数的代数性质、几何性质及解析
性质，根据下面的要求对矩阵范数结论加以仔细的区别：看这些矩阵是否依据依据矩阵范数次可乘性公理。
在许多应用方面，一般的矩阵(未必为Hermite型的)特征值的估计和扰动结果(第6章)是比较重要的。我们给出了GerŠgorin区域
理论的详细论述及它的一些现代的细化和相关的图表理论概念。
详细考察了正定矩阵及包括不等式在内的应用(第7章)。讨论了极点和奇异值分解以及矩阵逼近问题的应用。
分量非负矩阵及正矩阵(第8章)在许多应用中都可见到，在这些应用中必然会出现非负的量，如概率论，经济学，工程学等，其中的著名理论反映了这
些应用。我们在按照基本的步骤进行以范数应用为基础的非负矩阵、正矩阵、不可约矩阵的理论研究。
在下一卷，我们讨论了进一步的话题：值域及其推广；惯性矩阵，稳定矩阵，M-矩阵及相关的特殊分类；矩阵方程，Kronecker
积，Hadamard积；各种用于函数及矩阵的方法。
通过选择适合于特殊读者的章节，本书为一或两个学期的课程提供了基础。我们建议在教授本书时，教师可有选择的对本书各节内容进行讲授，其中包括第我章，
第2 章、第3章的大部分，第4章、第5章的Hermite矩阵及Hermite范数。
本书大部分章节都包含一些相对专业的、非传统的内容。例如，第2 章不仅包含关于单一矩阵酉三对角化的Schur基本定理，也讨论了一族矩阵的
同步三对角化。基本知识点通常安排在每章的前半部分，而更多的细节安排在每章的后半部分。这种安排方式有利于按照先后顺序介绍知识，并且能提高本书作为
参考书的作用。这样做同样也为授课都是提供了更多的选择。
对于其他域或更广泛代数集上的矩阵而言，许多讨论的结论都成立或经过推广后能够成立。然而，我们有意地将讨论的范围限定于实数和复数域，其中我们使用了
和形式代数方法一样的经典的分析方法。
虽然我们认为矩阵是复矩阵，但大部分的例子都只限定在实数范围内，没有更深入地去讨论复数矩阵的情形。熟悉复数算术对于理解矩阵分析是必要的，
这方面的知识安排在附录中。其他具有概括性的附录包含了几个外围的但非核心的话题，如Weierstrass定理及凸性。
我们给出了许多习题及问题，因为我们觉得对于理解矩阵分析及其应用来说，它们是非常重要的。练习题在每个小节中给出，一般都比较简单，是本小节
概念的直接应用。我们建议读者最好选择其中之一来做。问题通常在每节的结尾(不按特别的次序)列出，它们涵盖了本节的难点和(由理论到计算的)类型，展
开了主题，推广了特殊的知识点，或者暗示了主要思想的不同证明。有一定难度的问题我们都给了重要的暗示。一些问题的结论在其他问题或在教材本身中都有涉
及。我们觉得读者练习及解答问题是比较重要的。
本书附录后列出的参考文献为那些希望看到矩阵分析不同论述的读者提供了额外的信息。本书用易简短的易记符号来引用这些参考书，如Jones和
Smith写的一本书被记为[JSm]。作者将这些参考文献按照字母顺序来进行排列。
参考文献并不详尽。在这样一个多主题书籍中，为了节省有限的篇幅，我们将教材中的引文数量压缩到了最小。我们将诸如那些我们已经明确使用的论文
安排在了大部分章节后的简短讨论中，但是我们没有去搜集历史上的关于经典结论的参考文献。大量的参考书目由我们所参考的更专业的书籍给出了。读者也应该
知道广泛的当前的参考资料，这些资料涵盖了矩阵分析的各个方面，如《数值线性代数的国际标准题内关键词索引》[CaLe]及《数学评论》的第15
节、65节。
对于我们同事及学生的帮助性的建议，我们表示感激。这些学生花了大量的时间去整理课堂笔记及原始手稿，这些都构成了本书的原稿。他们是：Wayne
Barrett，Leroy Beasley，Bryan Cain， David Carlson， Dipa Choudhury，
Risana Chowdhury，Yoo Pyo Hong，Dmitry Krass，Dale Olesky， Stephen
Pierce，Leiba Rodman和Pauline van den Driessche。
guoping feng
Dec 6, 2007, 1:53:35 AM12/6/07
to dren...@googlegroups.com
good!
Reply all
Reply to author
Forward
0 new messages