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642
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
7.11 MINIMUM POLYNOMIALS AND KRYLOV METHODS
The characteristic polynomial plays a central role in the theoretical development
of linear algebra and matrix analysis, but it is not alone in this respect. There
are other polynomials that occur naturally, and the purpose of this section is to
explore some of them.
In this section it is convenient to consider the characteristic polynomial of
A ∈ C n×n to be c(x) = det (xI − A). This diﬀers from the deﬁnition given on
p. 492 only in the sense that the coeﬃcients of c(x) = det (xI − A) have diﬀerent
signs than the coeﬃcients of c(x) = det (A − xI). In particular, c(x) is a monic
ˆ
polynomial (i.e., its leading coeﬃcient is 1), whereas the leading coeﬃcient of c(x)
ˆ
is (−1)n . (Of course, the roots of c and c are identical.)
ˆ
Monic polynomials p(x) such that p(A) = 0 are said to be annihilating
polynomials for A. For example, the Cayley–Hamilton theorem (pp. 509, 532)
guarantees that c(x) is an annihilating polynomial of degree n. D
E T
H Minimum Polynomial for a Matrix IG
R There is a unique annihilating polynomial for A of minimal degree, and
this polynomial, denoted by m(x), is called the minimum polynomial
for A. The Cayley–Hamilton theorem guarantees that deg[m(x)] ≤ n. Y
P Proof. Only uniqueness needs to be proven. Let k be the smallest degree of
any annihilating polynomial for A. There is a unique annihilating polynomial
for A of degree k because if there were two diﬀerent annihilating polynomials
p1 (x) and p2 (x) of degree k, then d(x) = p1 (x) − p2 (x) would be a nonzero
polynomial such that d(A) = 0 and deg[d(x)] < k. Dividing d(x) by its leading
coeﬃcient would produce an annihilating polynomial of degree less than k, the
minimal degree, and this is impossible. O
C The ﬁrst problem is to describe what the minimum polynomial m(x) for
A ∈ C...

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