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Unformatted text preview: 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 meyer@ncsu.edu Buy from AMAZON.com 642 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 7.11 MINIMUM POLYNOMIALS AND KRYLOV METHODS The characteristic polynomial plays a central role in the theoretical development of linear algebra and matrix analysis, but it is not alone in this respect. There are other polynomials that occur naturally, and the purpose of this section is to explore some of them. In this section it is convenient to consider the characteristic polynomial of A ∈ C n×n to be c(x) = det (xI − A). This differs from the definition given on p. 492 only in the sense that the coefficients of c(x) = det (xI − A) have different signs than the coefficients of c(x) = det (A − xI). In particular, c(x) is a monic ˆ polynomial (i.e., its leading coefficient is 1), whereas the leading coefficient of c(x) ˆ is (−1)n . (Of course, the roots of c and c are identical.) ˆ Monic polynomials p(x) such that p(A) = 0 are said to be annihilating polynomials for A. For example, the Cayley–Hamilton theorem (pp. 509, 532) guarantees that c(x) is an annihilating polynomial of degree n. D E T H Minimum Polynomial for a Matrix IG R There is a unique annihilating polynomial for A of minimal degree, and this polynomial, denoted by m(x), is called the minimum polynomial for A. The Cayley–Hamilton theorem guarantees that deg[m(x)] ≤ n. Y P Proof. Only uniqueness needs to be proven. Let k be the smallest degree of any annihilating polynomial for A. There is a unique annihilating polynomial for A of degree k because if there were two different annihilating polynomials p1 (x) and p2 (x) of degree k, then d(x) = p1 (x) − p2 (x) would be a nonzero polynomial such that d(A) = 0 and deg[d(x)] < k. Dividing d(x) by its leading coefficient would produce an annihilating polynomial of degree less than k, the minimal degree, and this is impossible. O C The first problem is to describe what the minimum polynomial m(x) for A ∈ C...
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