This is a comprehensive research monograph intended for specialists in numerical linear algebra and related areas of research and application. It may also be suitable as the basis for a graduate course and certainly as a reference in matrix analysis and applied linear algebra.
The first two or three chapters can be read profitably by anyone wishing to understand the history, basic theory, and some applications of functions of matrices, perhaps as a supplement to the limited treatment provided by most books on linear algebra or the theory of differential equations. The waters become deeper soon after these early chapters. Higham’s 13-page survey (Chapter 11) in the Handbook of Linear Algebra is a useful introduction to the material in the monograph.
The foundation of the book rests on Higham’s development of the theory based on three equivalent definitions of f(A), where f is a scalar function and A is an n x n complex matrix: definitions based on the Jordan canonical form, on polynomial interpolation, and on the Cauchy integral formula (influenced by Nicholas Trefethen). The bulk of the book is devoted to numerical methods, their accuracy, stability, and computational cost.
Each chapter’s exposition is supplemented by a Notes and References section containing not only the expected historical comments and source publications, but also extra material. Each chapter has a list of problems — some labeled research problems — with solutions or references to solutions provide in Appendix E. Other appendices cover notation, linear algebra background (“Definitions and Useful Facts”), operation counts, and a description of the MATLAB Matrix Function Toolbox, which provides the implementation of key algorithms in the book.
Functions of Matrices concludes with a 625-item bibliography, prefaced by a graph describing the distribution of the year of publication of the references. Covering the years from 1850 to 2008, this chart makes clear the exponential growth in our understanding of matrix functions and related areas.
This work is unique, the only book devoted exclusively to matrix functions. This new book should join Higham’s Accuracy and Stability of Numerical Algorithms (Second Edition) as a classic of numerical and applied linear algebra.
Henry Ricardo (firstname.lastname@example.org) has retired from Medgar Evers College (CUNY) as Professor of Mathematics, but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text will be published in October by CRC Press.