*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **+ λx3
.
.
. Axp = xp−1 + λxp O
C 2
3 p In other words, the ﬁrst column x1 in P is a eigenvector for A associated with
λ. We already knew there had to be exactly one independent eigenvector for each
Jordan block because there are t = dim N (A − λI) Jordan blocks J (λ), but
now we know precisely where these eigenvectors are located in P.
Vectors x such that x ∈ N (A − λI)g but x ∈ N (A − λI)g−1 are called
generalized eigenvectors of order g associated with λ. So P consists of an
eigenvector followed by generalized eigenvectors of increasing order. Moreover,
the columns of P form a Jordan chain analogous to (7.7.2) on p. 576; i.e.,
p−i
xi = (A − λI)
xp implies P must have the form
P= p−1 (A − λI) p−2 xp | (A − λI) xp | · · · | (A − λI) xp | xp . (7.8.5) A complete set of Jordan chains associated with a given eigenvalue λ is determined in exactly the same way as Jordan chains for nilpotent matrices are Copyright c 2000 SIAM Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
594
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
determined except that the nested subspaces Mi deﬁned in (7.7.1) on p. 575
are redeﬁned to be
Mi = R (A − λI)i ∩ N (A − λI) for i = 0, 1, . . . , k, (7.8.6) where k = index (λ). Just as in the case of nilpotent matrices, it follows that
0 = Mk ⊆ Mk−1 ⊆ · · · ⊆ M0 = N (A − λI) (see Exercise 7.8.8). Since
is a nilpotent linear operator of index k (Example 5.10.4,
(A − λI)
k
/ It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu N ((AλI) ) p. 399), it can be argued that the same process used to build Jordan chains for
nilpotent matrices can be used to build Jordan chains for a general eigenvalue
λ. Below is a summary of the process adapted to the general case. D
E Constructing Jordan Chains For each λ ∈ σ (An×n ) , set Mi = R (A − λI)i ∩ N (A − λI) for
i = k − 1, k − 2,...

View
Full
Document