| 昨天偶然在图书馆看到了这书，感觉封面好像变了，翻翻才知道是2013 年刚出的 second edition,。于是把整本书浏览了一下，发现增加的，改动的地方真不少。这书实在是太给力了。比如增加了矩阵的低阶秩扰动后特征根的变化这类问题，专门另开了一节讲 矩阵的CS分解，还有 Weyr canonical form这种以前从未听过的分解。第0章中的预备知识也扩写了不少，而且我发现正文中的例子变得更加丰富，有些以前放在习题里的现在作为例子出现。习题作为一个重要的组成部分，作者也将习题扩充至1100多道，而且书后面绝大部分给出了提示，而且附录F给出了矩阵偶的标准型表，真是煞费苦心。下面是前言中的部分。 A comprehensive index is essential for a book that is intended for sustained use asa reference after initial use as a text. The index to the first edition had about 1,200 entries; the current index has more than 3,500 entries. An unfamiliar term encountered in the text should be looked up in the index, where a pointer to a definition (in Chapter 0 or elsewhere) is likely to be found。 New discoveries since 1985 have shaped the presentation of many topics and have stimulated inclusion of some new ones. A few examples of the latter are the Jordan canonical form of a rank-one perturbation, motivated by enduring student interest in the Google matrix; a generalization of real normal matrices (normal matrices A such that AA¯ is real); computable block matrix criteria for simultaneous unitary similarity or simultaneous unitary congruence; G. Belitskiis discovery that a matrix commutes with a Weyr canonical form if and only if it is block upper triangular and has a special structure; the discovery by K. C. O’ Meara and C. Vinsonhaler that, unlike the corresponding situation for the Jordan canonical form, a commuting family can be simultaneously upper triangularized by similarity in such a way that any one specified matrix in the family is in Weyr canonical form; and canonical forms for congruence and ∗congruence Queries from many readers have motivated changes in the way that some topics are presented. For example, discussion of Lidskii’s eigenvalue majorization inequalities was moved from a section primarily devoted to singular value inequalities to the section where majorization is discussed. Fortunately, a splendid new proof of Lidskii’s inequalities by C. K. Li and R. Mathias became available and was perfectly aligned with Chapter 4’s new approach to eigenvalue inequalities for Hermitian matrices. A second example is a new proof of Birkhoff’s theorem, which has a very different flavor from the proof in the first edition. Instructors accustomed to the order of topics in the first edition may be interested in a chapter-by-chapter summary of what is different in the new edition: 0. Chapter 0 has been expanded by about 75% to include a more comprehensive summary of useful concepts and facts. It is intended to serve as an as-needed reference. Definitions of terms and notations used throughout the book can be found here, but it has no exercises or problems. Formal courses and reading for self-study typically begin with Chapter 1. 1. Chapter 1 contains new examples related to similarity and the characteristic polynomial, as well as an enhanced emphasis on the role of left eigenvectors in matrix analysis 2. Chapter 2 contains a detailed presentation of real orthogonal similarity, an exposition of McCoy’s theorem on simultaneous triangularization, and a rigorous treatment of continuity of eigenvalues that makes essential use of both the unitary and triangular aspects of Schur’s unitary triangularization theorem. Section 2.4 (Consequences of Schur’s triangularization theorem) is almost twice the length of the corresponding section in the first edition. There are two new sections, one devoted to the singular value decomposition and one devoted to the C S decomposition. Early introduction of the singular value decomposition permits this essential tool of matrix analysis to be used throughout the rest of the book 3. Chapter 3 approaches the Jordan canonical form via the Weyr characteristic; it contains an exposition of the Weyr canonical form and its unitary variant that were not in the first edition. Section 3.2 (Consequences of the Jordan canonical form) discusses many new applications; it contains 60% more material than the corresponding section in the first edition. 4. Chapter 4 now has a modern presentation of variational principles and eigenvalue inequalities for Hermitian matrices via subspace intersections. It contains an expanded treatment of inverse problems associated with interlacing and other classical results. Its detailed treatment of unitary congruence includes Youla’s theorem (a normal form for a square complex matrix A under unitary congruence that is associated with the eigenstructure of AA¯), as well as canonical forms for conjugate normal, congruence normal, and squared normal matrices. It also has an exposition of recently discovered canonical forms for congruence and ∗ congruence and new algorithms to construct a basis of a coneigenspace. ............................ F. Appendix F tabulates a modern list of canonical forms for a pair of Hermitian matrices, or a pair of matrices, one of which is symmetric and the other is skew symmetric. These canonical pairs are applications of the canonical forms for congruence and ∗ congruence presented in Chapter 4.
对于这部经典之作的再版，广大读者有福了。个人认为，无论你是数学哪个行业的，翻一翻总会受益良多。 对于考研的同学，这本书也是强烈推荐，正文中出现的绝大多数例子绝不会亚于很多学校考研试题中的压轴题，读完前几章对考研肯定是大有帮助的。 此书还未影印，所以一般买不到（亚马逊上原版400多）。 |
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