# Y p for j 1 to n execute the j th arnoldi step in

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: /obidos/ASIN/0898714540 645 Since 0 = p(A) = m(A)q (A) + r(A) = r(A), It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] it follows that r(x) = 0; otherwise r(x), when normalized to be monic, would be an annihilating polynomial having degree smaller than the degree of the minimum polynomial. The structure of the minimum polynomial for A is related to the diagonalizability of A. By combining the fact that kj = index (λj ) is the size of the largest Jordan block for λj with the fact that A is diagonalizable if and only if all Jordan blocks are 1 × 1, it follows that A is diagonalizable if and only if kj = 1 for each j, which, by (7.11.1), is equivalent to saying that m(x) = (x − λ1 )(x − λ2 ) · · · (x − λs ). In other words, A is diagonalizable if and only if its minimum polynomial is the product of distinct linear factors. Below is a summary of the preceding observations about properties of m(x). D E T H Properties of the Minimum Polynomial Let A ∈ C n×n with σ (A) = {λ1 , λ2 , . . . , λs } . IG R • The minimum polynomial of A is the unique monic polynomial m(x) of minimal degree such that m(A) = 0. • m(x) = (x − λ1 )k1 (x − λ2 )k2 · · · (x − λs )ks , where kj = index (λj ). • m(x) divides every polynomial p(x) such that p(A) = 0. In particular, m(x) divides the characteristic polynomial c(x). (7.11.4) • m(x) = c(x) if and only if geo mult (λj ) = 1 for each λj or, equivalently, alg mult (λj ) = index (λj ) for each j, in which case A is called a nonderogatory matrix. Y P O C • A is diagonalizable if and only if m(x) = (x − λ1 )(x − λ2 ) · · · (x − λs ) (i.e., if and only if m(x) is a product of distinct linear factors). The next immediate aim is to extend the concept of the minimum polynomial for a matrix to formulate the notion of a minimum polynomial for a vector. 86 To do so, it’s helpful to introduce Krylov sequences, subspaces, and matrices. 86 Aleksei Nikolaevich Krylov (1863–1945) showed in 1...
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