Unlike solving ax b the eigenvalue problem generally

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: possess complex eigenvalues and eigenvectors. O C As we have seen, computing eigenvalues boils down to solving a polynomial equation. But determining solutions to polynomial equations can be a formidable task. It was proven in the nineteenth century that it’s impossible to express the roots of a general polynomial of degree five or higher using radicals of the coefficients. This means that there does not exist a generalized version of the quadratic formula for polynomials of degree greater than four, and general polynomial equations cannot be solved by a finite number of arithmetic operations n involving +,−,×,÷, √ . Unlike solving Ax = b, the eigenvalue problem generally requires an infinite algorithm, so all practical eigenvalue computations are accomplished by iterative methods—some are discussed later. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Buy from AMAZON.com 494 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 For theoretical work, and for textbook-type problems, it’s helpful to express the characteristic equation in terms of the principal minors. Recall that an r × r principal submatrix of An×n is a submatrix that lies on the same set of r rows and columns, and an r × r principal minor is the determinant of an r × r principal submatrix. In other words, r × r principal minors are obtained by deleting the same set of n−r rows and columns, and there are n = n!/r!(n−r)! r such minors. For example, the 1 × 1 principal minors of ⎛ ⎞ −3 1 −3 A = ⎝ 20 3 10 ⎠ (7.1.4) 2 −2 4 D E are the diagonal entries −3, 3, and 4. The 2 × 2 principal minors are −3 20 1 = −29, 3 −3 2 −3 = −6, 4 and 3 −2 10 = 32, 4 T H and the only 3 × 3 principal minor is det (A) = −18. Related to the principal minors are the symmetric functions of the eigenvalues. The k th symmetric function of λ1 , λ2 , . . . , λn is defined to be the sum of the product of the eigenvalues taken k at a time. T...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online