# akk where at the j th step of the process the vector

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Schmidt works ﬁne to produce m(x) in exact arithmetic, things are not so nice in ﬂoating-point arithmetic. For example, if A has a dominant eigenvalue, then, as explained in the power method (Example 7.3.7, p. 533), Ak asymptotically approaches the dominant spectral projector G1 , so, as k grows, Ak becomes increasingly close to span I, A, . . . , Ak−1 . Consequently, ﬁnding the ﬁrst Ak that is truly in span I, A, . . . , Ak−1 is an ill-conditioned problem, and Gram–Schmidt may not work well in ﬂoatingpoint arithmetic—the modiﬁed Gram–Schmidt algorithm (p. 316), or a version of Householder reduction (p. 341), or Arnoldi’s method (p. 653) works better. Fortunately, explicit knowledge of the minimum polynomial often is not needed in applied work. T H IG R The relationship between the characteristic polynomial c(x) and the minimum polynomial m(x) for A is now transparent. Since Y P c(x) = (x − λ1 )a1 (x − λ2 )a2 · · · (x − λs )as , where aj = alg mult (λj ), m(x) = (x − λ1 )k1 (x − λ2 )k2 · · · (x − λs )ks , where kj = index (λj ), and O C it’s clear that m(x) divides c(x). Furthermore, m(x) = c(x) if and only if alg mult (λj ) = index (λj ) for each λj ∈ σ (A) . Matrices for which m(x) = c(x) are said to be nonderogatory matrices, and they are precisely the ones for which geo mult (λj ) = 1 for each eigenvalue λj because m(x) = c(x) ⇐⇒ alg mult (λj ) = index (λj ) for each j ⇐⇒ there is only one Jordan block for each λj ⇐⇒ there is only one independent eigenvector for each λj ⇐⇒ geo mult (λj ) = 1 for each λj . In addition to dividing the characteristic polynomial c(x), the minimum polynomial m(x) divides all other annihilating polynomials p(x) for A because deg[m(x)] ≤ deg[p(x)] insures the existence of polynomials q (x) and r(x) (quotient and remainder) such that p(x) = m(x)q (x) + r(x), Copyright c 2000 SIAM where deg[r(x)] &lt; deg[m(x)]. Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.11 Minimum Polynomials and Krylov Methods http://www.amazon.com/exec...
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