In 1965 James Hardy Wilkinson published his classic work on numerical linear algebra, *The Algebraic Eigenvalue Problem*. His purpose was to survey the computational state of the art, but a significant part of the book was devoted to the mathematical foundations, some of which had been developed by Wilkinson himself. This happy mixture of the theoretical and computational has continued to characterize the area—computational advances go hand in hand with deeper mathematical understanding.

The purpose of this chapter is to provide the mathematical background for the first half of this volume. The first section consists of an exposition of the classic algebraic theory of eigensystems. The predominant theme of this section is the systematic reduction of matrices to simpler forms by similarity transformations. One of these forms — the Schur decomposition — is the goal of the QR algorithm, the workhorse of dense eigenvalue computations.

The second and third sections are devoted to the analysis of eigensystems. The second section concerns norms and the their relation to the spectrum of a matrix. Here we will pay special attention to the asymptotic behavior of powers of matrices, which are important in the analysis of many matrix algorithms. The third section is devoted to perturbation theory — how eigensystems behave when the original matrix is perturbed. This is a large subject, and we will only be able to survey the basics.

The majority of eigenvalue problems involve real matrices, and their reality often results in computational savings. However, the theory of eigensystems is best treated in terms of complex matrices. Consequently, throughout this chapter:

*Unless otherwise stated A is a complex matrix of order n*.

1. The Algebra of Eigensystems

In this section we will develop the classical theory of eigenvalues, eigenvectors, and reduction by similarity transformations, with an emphasis on the reduction of a matrix by similarity transformations to a simpler form. We begin with basic definitions and their consequences.