7.5 EIGENVALUE PROBLEMS

Problems involving the determination of eigenvalues and eigenvectors often arise in science and engineering. For example, eigenvalues arise in connection with vibration problems in mechanical engineering, in discussing the stability of an aircraft in aeronautical engineering and in quantum mechanics. In this section, we study the problems of calculating the eigenvalues of a square matrix. The general problem of finding all eigenvalues of a non-symmetric matrix is much more difficult as it easily leads to stability problems with respect to perturbations. Most of the algorithms for estimating eigenvalues of a symmetric matrix make use of similarity transformations and are therefore carried out in two stages. In the first stage, the matrix is reduced to a suitable form and, in the second stage, the method of determining the eigenvalues is executed. Before proceeding with the method, we briefly summarize the basic results and definitions.

We say that a matrix is similar to another matrix if there exists a non-singular matrix such that

(7.5 1)

If ? is an eigenvalue of and x is the corresponding eigenvector

(7.5 2)

Suppose that matrix is similar to . If ? is an eigenvalue of with associated eigenvector x, then ? is also an eigenvalue of with x as the associated eigenvector of .

If is triangular or diagonal, the diagonal entries of are the eigenvalues of .

A real matrix is orthogonal iff