TABLE OF CONTENTS Eigenvalue problems are particularly interesting in scientific computing, because the best algorithms for finding eigenvalues are powerful, yet particularly far from obvious. Here, we review the mathematics of eigenvalues and eigenvectors. Algorithms are discussed in later lectures.
Let 
be a square matrix. A nonzero vector 
is an eigenvector of A, and 
is its corresponding eigenvalue, if
The idea here is that the action of a matrix A on a subspace S of 
may sometimes mimic scalar multiplication. When this happens, the special subspace S is called an eigenspace, and any nonzero x ? S is an eigenvector.
The set of all the eigenvalues of a matrix A is the spectrum of A, a subset of 
denoted by ?( A).
Eigenvalue problems have a very different character from the problems involving square or rectangular linear systems of equations discussed in the previous lectures. For a system of equations, the domain of A could be one space and the range could be a different one. In Example 1.1, for example, A mapped n-vectors of polynomial coefficients to m-vectors of...
