**Unformatted text preview: **} must be a Jordan chain such
that (A − I)x1 = 0 and (A − I)x2 = x1 , while x3 is another eigenvector (not
dependent on x1 ). Suppose we try to build the Jordan chains in P by starting
with the eigenvectors
⎛
⎞
⎛⎞
1
0
x1 = ⎝ 0 ⎠
and x3 = ⎝ 1 ⎠
(7.8.7)
−2
0 D
E T
H IG
R obtained by solving (A − I)x = 0 with straightforward Gauss–Jordan elimination. This naive approach fails because x1 ∈ R (A − I) means (A − I)x2 = x1 is
an inconsistent system, so x2 cannot be determined. Similarly, x3 ∈ R (A − I)
insures that the same diﬃculty occurs if x3 is used in place of x1 . In other
words, even though the vectors in (7.8.7) constitute an otherwise perfectly good
basis for N (A − I), they are not suitable for building Jordan chains. You are
asked in Exercise 7.8.2 to ﬁnd the correct basis for N (A − I) that will yield the
Jordan chains that constitute P. Y
P Example 7.8.4 O
C Problem: What do the results concerning the Jordan form for A ∈ C n×n say
about the decomposition of C n into invariant subspaces?
Solution: Consider P−1 AP = J = diag (J(λ1 ), J(λ2 ), . . . , J(λs )) , where the
J(λj ) ’s are the Jordan segments and P = P1 | P2 | · · · | Ps is a matrix of
Jordan chains as described in (7.8.5) and on p. 594. If A is considered as a
linear operator on C n , and if the set of columns in Pi is denoted by Ji , then
the results in §4.9 (p. 259) concerning invariant subspaces together with those
in §5.9 (p. 383) about direct sum decompositions guarantee that each R (Pi ) is
an invariant subspace for A such that
C n = R (P1 ) ⊕ R (P2 ) ⊕ · · · ⊕ R (Ps ) and J(λi ) = A/
R(Pi ) Ji . More can be said. If alg mult (λi ) = mi and index (λi ) = ki , then Ji is a
linearly independent set containing mi vectors, and the discussion surrounding Copyright c 2000 SIAM Buy online from SIAM
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596
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
(7.8.5) insures that each column in Ji belongs to N (A ...

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