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**Unformatted text preview: **Schmidt works ﬁne to produce m(x) in exact arithmetic, things are not so nice in ﬂoating-point arithmetic. For example, if A
has a dominant eigenvalue, then, as explained in the power method (Example
7.3.7, p. 533), Ak asymptotically approaches the dominant spectral projector
G1 , so, as k grows, Ak becomes increasingly close to span I, A, . . . , Ak−1 .
Consequently, ﬁnding the ﬁrst Ak that is truly in span I, A, . . . , Ak−1 is
an ill-conditioned problem, and Gram–Schmidt may not work well in ﬂoatingpoint arithmetic—the modiﬁed Gram–Schmidt algorithm (p. 316), or a version
of Householder reduction (p. 341), or Arnoldi’s method (p. 653) works better.
Fortunately, explicit knowledge of the minimum polynomial often is not needed
in applied work. T
H IG
R The relationship between the characteristic polynomial c(x) and the minimum polynomial m(x) for A is now transparent. Since Y
P c(x) = (x − λ1 )a1 (x − λ2 )a2 · · · (x − λs )as , where aj = alg mult (λj ), m(x) = (x − λ1 )k1 (x − λ2 )k2 · · · (x − λs )ks , where kj = index (λj ), and O
C it’s clear that m(x) divides c(x). Furthermore, m(x) = c(x) if and only if
alg mult (λj ) = index (λj ) for each λj ∈ σ (A) . Matrices for which m(x) = c(x)
are said to be nonderogatory matrices, and they are precisely the ones for
which geo mult (λj ) = 1 for each eigenvalue λj because
m(x) = c(x) ⇐⇒ alg mult (λj ) = index (λj ) for each j
⇐⇒ there is only one Jordan block for each λj
⇐⇒ there is only one independent eigenvector for each λj
⇐⇒ geo mult (λj ) = 1 for each λj . In addition to dividing the characteristic polynomial c(x), the minimum
polynomial m(x) divides all other annihilating polynomials p(x) for A because deg[m(x)] ≤ deg[p(x)] insures the existence of polynomials q (x) and
r(x) (quotient and remainder) such that
p(x) = m(x)q (x) + r(x), Copyright c 2000 SIAM where deg[r(x)] < deg[m(x)]. Buy online from SIAM
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7.11 Minimum Polynomials and Krylov Methods
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