The basic structure of the first edition has been preserved in the second because itremains congruent with the goal of writing “a book that would be a useful moderntreatment of a broad range of topics. . . [that] may be used as an undergraduate orgraduate text and as a self-contained reference for a variety of audiences.” The quotationis from the Preface to the First Edition, whose declaration of goals for the work remainsunchanged.

What is different in the second edition?

The core role of canonical forms has been expanded as a unifying element inunderstanding similarity (complex, real, and simultaneous), unitary equivalence, unitarysimilarity, congruence, *congruence, unitary congruence, triangular equivalence,and other equivalence relations. More attention is paid to cases of equality in the manyinequalities considered in the book. Block matrices are a ubiquitous feature of theexposition in the new edition.

Learning mathematics has never been a spectator sport, so the new edition continuesto emphasize the value of exercises and problems for the active reader. Numerous2-by-2 examples illustrate concepts throughout the book. Problem threads (some spanseveral chapters) develop special topics as the foundation for them evolves in the text.For example, there are threads involving the adjugate matrix, the compound matrix,finite-dimensional quantum systems, the Loewner ellipsoid and the Loewner–Johnmatrix, and normalizable matrices; see the index for page references for these threads.The first edition had about 690 problems; the second edition has more than 1,100.Many problems have hints; they may be found in an appendix that appears just beforethe index.

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 1,200entries; the current index has more than 3,500 entries. An unfamiliar term encounteredin the text should be looked up in the index, where a pointer to a definition (in Chapter0 or elsewhere) is likely to be found.

New discoveries since 1985 have shaped the presentation of many topics and havestimulated 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 inthe Google matrix; a generalization of real normal matrices (normal matrices A suchthat A ¯A is real); computable block matrix criteria for simultaneous unitary similarityor simultaneous unitary congruence; G. Belitskii’s discovery that a matrix commuteswith a Weyr canonical form if and only if it is block upper triangular and has aspecial structure; the discovery by K. C. O’Meara and C. Vinsonhaler that, unlike thecorresponding situation for the Jordan canonical form, a commuting family can besimultaneously upper triangularized by similarity in such a way that any one specifiedmatrix in the family is in Weyr canonical form; and canonical forms for congruenceand ＊congruence.

Queries from many readers have motivated changes in the way that some topics arepresented. For example, discussion of Lidskii’s eigenvalue majorization inequalitieswas moved from a section primarily devoted to singular value inequalities to thesection where majorization is discussed. Fortunately, a splendid new proof of Lidskii’sinequalities by C. K. Li and R. Mathias became available and was perfectly alignedwith Chapter 4’s new approach to eigenvalue inequalities for Hermitian matrices. Asecond example is a new proof of Birkhoff’s theorem, which has a very different flavorfrom the proof in the first edition.

Instructors accustomed to the order of topics in the first edition may be interested ina chapter-by-chapter summary of what is different in the new edition:

Chapter 0 has been expanded by about 75% to include a more comprehensivesummary of useful concepts and facts. It is intended to serve as an as-neededreference. Definitions of terms and notations used throughout the book can befound here, but it has no exercises or problems. Formal courses and reading forself-study typically begin with Chapter 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 matrixanalysis.

Chapter 2 contains a detailed presentation of real orthogonal similarity, anexposition of McCoy’s theorem on simultaneous triangularization, and a rigoroustreatment of continuity of eigenvalues that makes essential use of both theunitary and triangular aspects of Schur’s unitary triangularization theorem. Section2.4 (Consequences of Schur’s triangularization theorem) is almost twice thelength 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 CS decomposition.Early introduction of the singular value decomposition permits thisessential tool of matrix analysis to be used throughout the rest of the book.

Chapter 3 approaches the Jordan canonical form via the Weyr characteristic; itcontains an exposition of the Weyr canonical form and its unitary variant thatwere not in the first edition. Section 3.2 (Consequences of the Jordan canonicalform) discusses many new applications; it contains 60% more material than thecorresponding section in the first edition.

Chapter 4 now has a modern presentation of variational principles and eigenvalueinequalities for Hermitian matrices via subspace intersections. It containsan expanded treatment of inverse problems associated with interlacing and other classical results. Its detailed treatment of unitary congruence includes Youla’stheorem (a normal form for a square complex matrix A under unitary congruencethat is associated with the eigenstructure of A ¯A), as well as canonical forms forconjugate normal, congruence normal, and squared normal matrices. It also has anexposition of recently discovered canonical forms for congruence and ∗congruenceand new algorithms to construct a basis of a coneigenspace.

Chapter 5 contains an expanded discussion of norm duality, many new problems,and a treatment of semi-inner products that finds application in a discussion offinite-dimensional quantum systems in Chapter 7.

Chapter 6 has a newtreatment of the “disjoint discs” aspect of Gerˇsgorin’s theoremand a reorganized discussion of eigenvalue perturbations, including differentiabilityof a simple eigenvalue.

Chapter 7 has been reorganized now that the singular value decomposition isintroduced in Chapter 2. There is a new treatment of the polar decomposition, newfactorizations related to the singular value decomposition, and special emphasis onrowand column inclusion. The von Neumann trace theorem (proved via Birkhoff’stheorem) is now the foundation on which many applications of the singular valuedecomposition are built. The Loewner partial order and block matrices are treatedin detail with new techniques, as are the classical determinant inequalities forpositive definite matrices.

Chapter 8 uses facts about left eigenvectors developed in Chapter 1 to streamline itsexposition of the Perron–Frobenius theory of positive and nonnegative matrices.

D. Appendix D contains new explicit perturbation bounds for the zeroes of a polynomialand the eigenvalues of a matrix.

F. Appendix F tabulates a modern list of canonical forms for a pair of Hermitianmatrices, or a pair of matrices, one of which is symmetric and the other is skewsymmetric. These canonical pairs are applications of the canonical forms forcongruence and ∗congruence presented in Chapter 4.

Readers who are curious about the technology of book making may be interestedto know that this book began as a set of LATEX files created manually by a company inIndia from hard copy of the first edition. Those files were edited and revised using theScientific WorkPlace¬R graphical user interface and typesetting system.

The cover art for the second edition was the result of a lucky encounter on a Deltaflight from Salt Lake City to Los Angeles in spring 2003. The young man in themiddle seat said he was an artist who paints abstract paintings that are sometimesmathematically inspired. In the course of friendly conversation, he revealed that hisspecial area of mathematical enjoyment was linear algebra, and that he had studiedMatrix Analysis. After mutual expressions of surprise at the chance nature of ourmeeting, and a pleasant discussion, we agreed that appropriate cover art would enhancethe visual appeal of the second edition; he said he would send something to consider.In due course a packet arrived from Seattle. It contained a letter and a stunning 4.5- by5-inch color photograph, identified on the back as an image of a 72- by 66-inch oil oncanvas, painted in 2002. The letter said that “the painting is entitled Surprised Againon the Diagonal and is inspired by the recurring prevalence of the diagonal in mathwhether it be in geometry, analysis, algebra, set theory or logic. I think that it would be an attractive addition to your wonderful book.” Thank you, Lun-Yi Tsai, for yourwonderful cover art!

A great many students, instructors, and professional colleagues have contributedto the evolution of this new edition since its predecessor appeared in 1985. Specialthanks are hereby acknowledged to T. Ando, Wayne Barrett, Ignat Domanov, Jim Fill,Carlos Martins da Fonseca, Tatiana Gerasimova, Geoffrey Goodson, Robert Guralnick,Thomas Hawkins, Eugene Herman, Khakim Ikramov, Ilse Ipsen, Dennis C. Jespersen,Hideki Kosaki, Zhongshan Li, Teck C. Lim, Ross A. Lippert, Roy Mathias, DennisMerino, Arnold Neumaier, Kevin O’Meara, Peter Rosenthal, Vladimir Sergeichuk,Wasin So, Hugo Woerdeman, and Fuzhen Zhang.

R.A.H.