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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
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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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