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# Consequently numerical computation of jordan forms is

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Unformatted text preview: from AMAZON.com 7.7 Nilpotent Matrices and Jordan Structure http://www.amazon.com/exec/obidos/ASIN/0898714540 585 invariant subspace for L, and the matrix representation of L with respect to the basis J = Jb1 ∪ Jb2 ∪ · · · ∪ Jbt is ⎞ ⎛ N1 0 · · · 0 ⎜ 0 N2 · · · 0 ⎟ [L]J = ⎜ . (7.7.8) . . ⎟ in which Nj = L V .. ⎝. / j Jb . .⎠ . . . . . 0 0 j · · · Nt It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Exercises for section 7.7 D E 7.7.1. Can the index of an n × n nilpotent matrix ever exceed n? 7.7.2. Determine all possible Jordan forms N for a 4 × 4 nilpotent matrix. T H 7.7.3. Explain why the number of blocks of size i × i or larger in the Jordan form for a nilpotent matrix is given by rank Li−1 − rank Li . 7.7.4. For a nilpotent matrix L of index k, let Mi = R Li ∩ N (L). Prove that Mi ⊆ Mi−1 for each i = 0, 1, . . . , k. IG R 7.7.5. Prove that R Lk−1 ∩ N (L) = R Lk−1 for all nilpotent matrices L of index k > 1. In other words, prove Mk−1 = R Lk−1 . 7.7.6. Let L be a nilpotent matrix of index k > 1. Prove that if the columns of B are a basis for R Li for i ≤ k − 1, and if {v1 , v2 , . . . , vs } is a basis for N (LB), then {Bv1 , Bv2 , . . . , Bvs } is a basis for Mi . Y P 7.7.7. Find P and N such that P−1 LP = N is in Jordan form, where ⎛ ⎞ 3 3 2 1 ⎜ −2 −1 −1 −1 ⎟ L=⎝ ⎠. 1 −1 0 1 −5 −4 −3 −2 O C 7.7.8. Determine the Jordan form for the following ⎛ 41 30 15 7 4 ⎜ −54 −39 −19 −9 −6 ⎜ 6 2 1 2 ⎜9 ⎜ −3 −2 1 ⎜ −6 −5 L=⎜ ⎜ −32 −24 −13 −6 −2 ⎜ −7 −2 0 −3 ⎜ −10 ⎝ −4 −3 −2 −1 0 17 12 6 3 2 Copyright c 2000 SIAM 8 × 8 nilpotent matrix. ⎞ 6 1 3 −8 −2 −4 ⎟ ⎟ 1 0 1⎟ ⎟ −1 0 0⎟ ⎟. −5 −1 −2 ⎟ ⎟ 0 3 −2 ⎟ ⎠ −1 −1 0 3 2 1 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 586 Chapter 7 Eigenvalues and Eigenv...
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