Introduction to Computational Linear Algebra

Chapter 5

Algorithms for the Eigenvalue

Problem

5.1 Basic Principles

5.1.1 Why Compute the Eigenvalues of a Square Matrix?

The eigenvalues of an n × n square matrix are canonical parameters of the

corresponding linear application L on a finite dimensional vector space E of

dimension n.

Given the application L, the matrix A depends on a reference basis B in the

vector space E, which we note by A = M

B

(L). When changing the reference

basis in E, the matrix A is changed into a matrix

˜

A that is similar to A, i.e.,

˜

A = X

−1

AX, where the columns of X = [x

1

, ··· , x

n

] are the vectors of the

new basis expressed in terms of the previous one. A natural inquiry consists ...