The Theory of Matrices in Numerical Analysis

This text, first published in 1964, is a classic work in numerical linear algebra. Its purpose is to develop the tools and analytical methods needed to evaluate computational techniques for solving systems of linear equations and finding eigenvalues. It is in no sense an introductory book. It assumes a decent background in linear algebra, and then it digs deeply into the analysis of linear systems right from the beginning.

Householder wrote this book in an era when people wrote much of their own software for numerical linear algebra. Householder’s work provided guidance and theoretical foundations, but he generally stayed away from any kind of implementation details. Because of the work of Householder and his colleagues, some very good software packages for numerical linear algebra have been developed over the years — so good that software engineers are now strongly encouraged not to write their own code. Householder’s name remains prominent, not least because of the “Householder transformation”, a reduction of an \(n \times n\) symmetric matrix to tridiagonal form using \(n – 2\) orthogonal transformations. It is also an essential step in the QR algorithm — decomposition of a real matrix into the product of an orthogonal matrix Q and an upper triangular matrix R.

The book begins with elementary matrices, projections, determinantal identities, orthogonal polynomials and the Lanczos algorithm for tridiagonalization. Chapter two develops the idea of matrix norms, a concept that is particularly useful in error analysis. The third chapter, the last of the preliminary work, takes up localization. Localization results pertain to eigenvalues and their presence or absence in regions of the complex plane. Associated with these are separation theorems for the eigenvalues of normal matrices.

The remainder of the book surveys the main methods for solving a linear system of equations, calculating a matrix inverse, and finding eigenvalues and eigenvectors. The emphasis throughout is on exploring the mathematical foundations that underlie these methods and the mathematical relationships between them.

The book has descriptions of methods, proofs, and plenty of exercises. But there is not a single numerical example, and very little of anything else that might be called an example. This makes the subject — one that it is intrinsically numerical — look very rarified.

This would not be a book of choice for learning numerical linear algebra. It gets too quickly into the details and would soon overwhelm someone new to the field. A good alternative might be G. W. Stewart’s Introduction to Matrix Computations.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.