Horn and Johnson’s 1985 book Matrix Analysis is the standard reference for the subject, along with the companion volume Topics in Matrix Analysis (1991). This second edition, published 28 years after the first, is long-awaited. It’s a major revision: 643 pages up from 561 and with much more on each page thanks to pages that are wider and taller. The number of problems and the number of index entries have both increased, by 60% and a factor 3, respectively, according to the preface. Hints for solutions of the problems are now given in an appendix.
The number of chapters is unchanged and their titles are essentially the same. New material has been added, such as the CS decomposition; existing material has been reorganized, with the singular value decomposition appearing much earlier now; and the roles of block matrices and left eigenvectors have been expanded.
Unlike the first edition, the book has been typeset in LaTeX (in Times Roman) and it’s been beautifully done, except for the too-large solid five-pointed star used in some displays. Moreover, the print quality is superb. Oddly, equations are not punctuated! (The same is true of the first edition, though I must admit I had not noticed.)
The new edition is clearly a must-have for anyone seriously interested in matrix analysis.
Note, however, that this book is not, and cannot be without greatly increasing its size, a comprehensive research monograph. Thus exhaustive references to the literature are not given (as stated in the preface to the original edition). Also, in some cases a story is partly told in the main text and completed in the Problems, or in the Notes and Further Reading. For example, Theorem 184.108.40.206 on page 184 compares the Jordan structure of the nonzero eigenvalues of AB and BA (previously a Problem in the first edition), but the comparison for zero eigenvalues is only mentioned in the Notes and Further Reading seven pages later and is not signposted in the main text.
The 37-page index is extremely comprehensive and covers the Problems as well as the main text. It’s not perfect: Sylvester equation is missing (or rather, is hidden as the subentry Sylvester’s theorem, linear matrix equations).
A final point: the References (bibliography) contains several books that are out of print from the indicated publisher but are available in reprints from other publishers, notably in the SIAM Classics in Applied Mathematics series. They are:
- Rajendra Bhatia, Perturbation Bounds for Matrix Eigenvalues, SIAM, 2007: hard copy, ebook.
- Françoise Chatelin, Eigenvalues of Matrices, SIAM, 2012: ebook, hard copy
- Charles Cullen, Matrices and Linear Transformations, Second edition, Dover, 1990: Google Books.
- Israel Gohberg, Peter Lancaster & Leiba Rodman, Matrix Polynomials, SIAM, 2009: hard copy, ebook.
- Israel Gohberg, Peter Lancaster & Leiba Rodman, Indefinite Linear Algebra and Applications, Birkhauser, 2005: ebook.
- Marvin Marcus & Henryk Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover, 1992: Google Books.
- Stephen Campbell & Carl Meyer, Generalized Inverses of Linear Transformations, SIAM, 2009: hard copy, ebook.