Evaluation of Eigenvalues and Eigenvectors

Before we discuss methods for computing eigenvalues, we recall a remark made in Chap. 4 . A given nth-degree polynomial p(c) is the characteristic polynomial of some matrix. The companion matrix of Eq. ( 3.247 ) is one such matrix. companion matrix Thus, given a general polynomial p, we can form a matrix A whose eigenvalues are the roots of the polynomial; and likewise, given a square matrix, we can write a polynomial in its eigenvalues. It is a well-known fact in the theory of equations that there is no general formula for the roots of a polynomial of degree greater than 4. This means that we cannot expect to have a direct method for calculating eigenvalues of any given matrix.