Computation of Eigenvectors and Eigenvalues and the Singular Value Decomposition

Abstract

Before we discuss methods for computing eigenvalues, we mention an interesting observation. Consider the polynomial, f(& λ),

$$ {{\lambda }^{p}} + {{a}_{{p - 1}}}{{\lambda }^{{p - 1}}} + ... + {{a}_{1}}\lambda + {{a}_{0}} $$

Now form the matrix, A,

$$ \left[ \begin{gathered} 0 1 0 ... 0 \hfill \\ 0 0 1 ... 0 \hfill \\ \ddots \hfill \\ 0 0 0 ... 0 \hfill \\ - {{a}_{0}} - {{a}_{1}} - {{a}_{2}} ... - {{a}_{{p - 1}}} \hfill \\ \end{gathered} \right] $$

The matrix A is called the companion matrix of the polynomial f. It is easy to see that the characteristic equation of A, equation (2.11) on page 68, is the polynomial f(λ):

$$ \det (A - \lambda I) = f(\lambda ) $$

Thus, given a general polynomial f, we can form a matrix A whose eigenvalues are the roots of the polynomial. 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; rather, we will have to use an iterative method.