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**Unformatted text preview: **herwise λ1 is another eigenvalue with the
same magnitude as λ1 . Consider f (z ) = (z/λ1 )n , and use the spectral representation (7.3.6) along with |λi /λ1 | < 1 for i = 2, 3, . . . , k to conclude that
A
λ1 Y
P n = f (A) = f (λ1 )G1 + f (λ2 )G2 + · · · + f (λk )Gk
= G1 + O
C λ2
λ1 n G2 + · · · + λk
λ1 (7.3.16) n Gk → G1 as n → ∞. Consequently, (An x0 /λn ) → G1 x0 ∈ N (A − λ1 I) for all x0 . So if
1
G1 x0 = 0 or, equivalently, x0 ∈ R (A − λ1 I), then An x0 /λn converges to an
/
1
eigenvector associated with λ1 . This means that the direction of An x0 tends
toward the direction of an eigenvector because λn acts only as a scaling factor
1
to keep the length of An x0 under control. Rather than using λn , we can scale
1
An x0 with something more convenient. For example, An x0 (for any vector
norm) is a reasonable scaling factor, but there are even better choices. For vectors
v, let m(v) denote the component of maximal magnitude, and if there is more
74 While the development of the power method was considered to be a great achievement when
R. von Mises introduced it in 1929, later algorithms relegated its computational role to that of
a special purpose technique. Nevertheless, it’s still an important idea because, in some way or
another, most practical algorithms for eigencomputations implicitly rely on the mathematical
essence of the power method. Copyright c 2000 SIAM Buy online from SIAM
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534
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
than one maximal component, let m(v) be the ﬁrst maximal component—e.g.,
m(1, 3, −2) = 3, and m(−3, 3, −2) = −3. It’s clear that m(αv) = αm(v) for
all scalars α. Suppose m(An x0 /λn ) → γ. Since (An /λn ) → G1 , we see that
1
1
An x0
(An /λn )x0
G1 x0
1
= lim
=
=x
n→∞ m(An x0 )
n→∞ m(An x0 /λn )
γ
1
lim It is illegal to print, duplicate, or distribute this material
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