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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 Buy from 596 Chapter 7 Eigenvalues and Eigenvectors (7.8.5) insures that each column in Ji belongs to N (A...
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