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# Since the distinct eigenvalues of a are a 1 2

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Unformatted text preview: ⎛ ⎞ ⎛ 000 010 0 N1 = ⎝ 0 0 0 ⎠ , N2 = ⎝ 0 0 0 ⎠ , N3 = ⎝ 0 000 000 0 T H 1 0 0 ⎞ 0 1⎠. 0 IG R Example 7.7.2 For a nilpotent matrix L, the theoretical development relies on a complicated basis for N (L) to derive the structure of the Jordan form N as well as the Jordan chains that constitute a nonsingular matrix P such that P−1 LP = N. But, after the dust settled, we saw that a basis for N (L) is not needed to construct N because N is completely determined simply by ranks of powers of L. A basis for N (L) is only required to construct the Jordan chains in P. Y P Question: For the purpose of constructing Jordan chains in P, can we use an arbitrary basis for N (L) instead of the complicated basis built from the Mi ’s? O C Answer: No! Consider the nilpotent matrix ⎛ ⎞ 20 1 L = ⎝ −4 0 −2 ⎠ and its Jordan form −4 0 −2 ⎛ 01 00 N=⎝ 0 0 ⎞ 0 0⎠. 0 If P−1 LP = N, where P = [ x1 | x2 | x3 ], then LP = PN implies that Lx1 = 0, Lx2 = x1 , and Lx3 = 0. In other words, B = {x1 , x3 } must be a basis for N (L), and Jx1 = {x1 , x2 } must be a Jordan chain built on top of x1 . If we try to construct such vectors by starting with the naive basis ⎛ ⎞ ⎛⎞ 1 0 x1 = ⎝ 0 ⎠ and x3 = ⎝ 1 ⎠ (7.7.7) −2 0 Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 582 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 for N (L) obtained by solving Lx = 0 with straightforward Gaussian elimination, we immediately hit a brick wall because x1 ∈ R (L) means Lx2 = x1 is an inconsistent system, so x2 cannot be determined. Similarly, x3 ∈ R (L) insures that the same diﬃculty occurs if x3 is used in place of x1 . In other words, even though the vectors in (7.7.7) constitute an otherwise perfectly good basis for N (L), they can’t be used to build P. It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Example 7.7.3 Example 7.7.4 Problem: Let Ln...
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