**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.
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To do so, it’s helpful to introduce Krylov sequences, subspaces, and matrices.
86 Aleksei Nikolaevich Krylov (1863–1945) showed in 1...

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