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Unformatted text preview: 931 how to use sequences of the form {b, Ab, A2 b, . . .} to construct the characteristic polynomial of a matrix (see Example 7.11.3 on p. 649). Krylov was a Russian applied mathematician whose scientific interests arose from his early training in naval science that involved the theories of buoyancy, stability, rolling and pitching, vibrations, and compass theories. Krylov served as the director of the Physics– Mathematics Institute of the Soviet Academy of Sciences from 1927 until 1932, and in 1943 he was awarded a “state prize” for his work on compass theory. Krylov was made a “hero of Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 646 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Krylov Sequences, Subspaces, and Matrices For A ∈ C n×n and 0 = b ∈ C n×1 , we adopt the following terminology. It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu • {b, Ab, A2 b, . . . , Aj −1 b} is called a Krylov sequence. • Kj = span b, Ab, . . . , Aj −1 b • Kn×j = b | Ab | · · · | Aj −1 b is called a Krylov matrix. is called a Krylov subspace. n×1 T H D E Since dim(Kj ) ≤ n (because Kj ⊆ C ), there is a first vector A b in the Krylov sequence that is a linear combination of preceding Krylov vectors. If k−1 k k−1 Ak b = αj Aj b, then we define v (x) = xk − IG R j =0 αj xj , j =0 and we say that v (x) is an annihilating polynomial for b relative to A because v (x) is a monic polynomial such that v (A)b = 0. The argument on p. 642 that establishes uniqueness of the minimum polynomial for matrices can be reapplied to prove that for each matrix–vector pair (A, b) there is a unique annihilating polynomial of b relative to A that has minimal degree. These observations are formalized below. Y P Minimum Polynomial for a Vector O C • The minimum polynomial for b ∈ C n×1 relative to A ∈ C n×n is defined to be the monic polynomial v (x) of minimal degree such that v (A)b = 0. • If Ak b is the first vector in the Krylov sequence {b, Ab, A3 b, . . .} that is a linear combinat...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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