The proof of uniqueness of the jordan form for a

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Unformatted text preview: ×n be a nilpotent matrix of index k. Provide an algorithm for constructing the Jordan chains that generate a nonsingular matrix P such that P−1 LP = N is in Jordan form. D E Solution: 1. Start with the fact that Mk−1 = R Lk−1 (Exercise 7.7.5), and determine a basis {y1 , y2 , . . . , yq } for R Lk−1 . T H 2. Extend {y1 , y2 , . . . , yq } to a basis for Mk−2 = R Lk−2 ∩ N (L) as follows. Find a basis {v1 , v2 , . . . , vs } for N (LB), where B is a matrix containing a basis for R Lk−2 —e.g., the basic columns of Lk−2 . The set {Bv1 , Bv2 , . . . , Bvs } is a basis for Mk−2 (see p. 211). Find the basic columns in [ y1 | y2 | · · · | yq | Bv1 | Bv2 | · · · | Bvs ]. Say they are {y1 , . . . , yq , Bvβ1 , . . . , Bvβj } (all of the yj ’s are basic because they are a leading linearly independent subset). This is a basis for Mk−2 that contains a basis for Mk−1 . In other words, IG R Y P Sk−1 = {y1 , y2 , . . . , yq } and Sk−2 = {Bvβ1 , Bvβ2 , . . . , Bvβj }. 3. Repeat the above procedure k − 1 times to construct a basis for N (L) that is of the form B = Sk−1 ∪ Sk−2 ∪ · · · ∪ S0 = {b1 , b2 , . . . , bt }, where Sk−1 ∪ Sk−2 ∪ · · · ∪ Si is a basis for Mi for each i = k − 1, k − 2, . . . , 0. 4. Build a Jordan chain on top of each bj ∈ B. If bj ∈ Si , then we solve Li xj = bj and set Jj = [ Li xj | Li−1 xj | · · · | Lxj | xj ]. The desired similarity transformation is Pn×n = [ J1 | J2 | · · · | Jt ]. O C Problem: Find P and N such that P−1 LP = N is in Jordan form, where ⎛ 1 ⎜3 ⎜ ⎜ −2 L=⎜ ⎜2 ⎝ −5 −3 Copyright c 2000 SIAM 1 1 −1 1 −3 −2 −2 5 0 0 −1 −1 0 1 0 0 −1 −1 ⎞ 1 −1 −1 3⎟ ⎟ −1 0⎟ ⎟. 1 0⎟ ⎠ −1 −1 0 −1 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.7 Nilpotent Matrices and Jordan Structure http://www.amazon.com/exec/obidos/ASIN/0898714540 583 Solution: First determine the Jordan form for L. Computing ri = rank Li reveals that r1 = 3, r2 = 1, and...
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