# Eigenvalues of matrices / Françoise Chatelin, with exercises by Mario Ahués and Françoise Chatelin

This book was originally published in two separate volumes by Mason, Paris: "Valeurs propres de matrices" (1988) and "Exercises de valeurs propres de matrices" (1989). This SIAM edition is a revised republication of the first english version of the book, published by John Wiley & Sons, Inc., in 1993.

The present book is a work on numerical analysis in depth whose mail goal is to give a modern and complete theory, on an elementary level, of the problem of calculation of eigenvalues of square matrices. The first chapter is devoted to develop the theory of linear operators between finite dimensional complex vector spaces involved in the eigenvalue problem (and much more!). It is remarkable the strongly computational proof of the existence of the Jordan form of the endomorphisms of ${\mathbb C}^n$ based on the Schur's form. The collection of problems and exercises of this chapter is spectacular.

Along the book the author uses the language of functional analysis, which has the effect of demonstrating the profound similarity between the different methods of approximation, while the systematic use of bases to represent invariant subspaces provides a geometric interpretation that enhances the traditional algebraic presentation of many algorithms in numerical matrix analysis.

Elements of spectral theory are studied in Chapter 2. If ${\rm sp}(A)$ denotes the set of eigenvalues of $A$, the detailed study of the singularities of the analytic function

$$

{\mathbb C}\setminus{\rm sp}(A)\to{\mathfrak M}_n({\mathbb C}),\, z\mapsto(A-zI)^{-1},

$$

called the "resolvent" of $A$, give rise to the so called Rellich-Kato and Rayleigh-Schrdingen expansions.

In Chapter 3 the author explains why it is interesting to compute eigenvalues: differential and difference equations, Markov chains, theory of economics, factorial analysis of data, the dynamics of structures, chemistry and Fredholm's integral equations, are the selected topics where the knowledge of eigenvalues or their approximations are crucial.

Error analysis is treated in Chapter 4. It includes a revision of the conditioning of a system and the crucial stability of a spectral problem, which open the doors to both the "a priori" analysis of errors and "a posteriori" analysis of errors.

Chapter 5 and 6 have a more technical character. They include, of course, the famous $QR$ and $QZ$ algorithms, and some numerical methods for "large" matrices: the Lanczos method, and the Arnoldi's method among others. Chapter 7 is devoted to Chebyshev's iterative methods, while polymorphic information processing with matrices is treated in Chapter 8. This includes the homotopic deviation, that is, the study of the analyticity of the map

$$

t\mapsto (A(t)-zI)^{-1}

$$

around $t=0$ and $t=\infty$.

This carefully written graduate-level book constitutes a useful reference for students and researchers in the field. It is a french book written in English language. Consequently, much of it is written in the definition-theorem-proof format, with its emphasis on the best currently available methods for a range of important problems. The bibliography is complemented by bibliographic comments at the end of each chapter. The author is a very recognized expert in the field, and so the selection of the included topics is very accurate.

**Submitted by Anonymous |

**7 / Feb / 2014