**Unformatted text preview: **. . , en } , and let vi (x) be the minimum polynomial
of ei with respect to C. Observe that v1 (x) = p(x) because Cej = ej +1 for
j = 1, . . . , n − 1, so
{e1 , Ce1 , C2 e1 , . . . , Cn−1 e1 } = {e1 , e2 , e3 , . . . , en } and
n−1 Cn e1 = Cen = C∗n = − n−1 αj ej +1 = −
j =0 αj Cj e1 =⇒ v1 (x) = p(x).
j =0 Since v1 (x) divides the LCM of all vi (x) ’s (which we know from (7.11.5) to be
the minimum polynomial m(x) for C ), we conclude that p(x) divides m(x).
But m(x) always divides p(x) —recall (7.11.4)—so m(x) = p(x). Copyright c 2000 SIAM Buy online from SIAM
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7.11 Minimum Polynomials and Krylov Methods
http://www.amazon.com/exec/obidos/ASIN/0898714540 649 Example 7.11.2 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] Poor Man’s Root Finder. The companion matrix is the source of what is
often called the poor man’s root ﬁnder because any general purpose algorithm
designed to compute eigenvalues (e.g., the QR iteration on p. 535) can be applied
to the companion matrix for a polynomial p(x) to compute the roots of p(x).
When used in conjunction with (7.1.12) on p. 497, the companion matrix is also
a poor man’s root bounder . For example, it follows that if λ is a root of p(x),
then
|λ| ≤ C ∞ = max{|α0 |, 1 + |α1 |, . . . , 1 + |αn−1 |} ≤ 1 + max |αi |. D
E The results on p. 647 insure that the minimum polynomial v (x) for every
nonzero vector b relative to A ∈ C n×n divides the minimum polynomial m(x)
for A, which in turn divides the characteristic polynomial c(x) for A, so it
follows that every v (x) divides c(x). This suggests that it might be possible to
construct c(x) as a product of vi (x) ’s. In fact, this is what Krylov did in 1931,
and the following example shows how he did it. T
H IG
R Example 7.11.3 Krylov’s method for constructing the characteristic polynomial for A ∈ C n×n
as a product of minimum polynomials for vectors is as follows.
k−1 Sta...

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