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Unlike solving ax b the eigenvalue problem generally

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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...
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