Whats important in the jordan structure of l is the

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Unformatted text preview: k-diagonal matrix ⎞ ⎛ N1 0 · · · 0 ⎜ 0 N2 · · · 0 ⎟ (7.7.6) P−1 LP = N = ⎜ . .. . ⎟ ⎝. . .⎠ . . Y P 0 0 · · · Nt in which each Nj is a nilpotent matrix having ones on the superdiagonal and zeros elsewhere—see (7.7.5). O C • The number of blocks in N is given by t = dim N (L). • • The size of the largest block in N is k × k. The number of i × i blocks in N is νi = ri−1 − 2ri + ri+1 , where ri = rank Li —this follows from (7.7.4). • If B = Sk−1 ∪ Sk−2 ∪ · · · ∪ S0 = {b1 , b2 , . . . , bt } is a basis for N (L) derived from the nested subspaces Mi = R Li ∩ N (L), then the set of vectors J = Jb1 ∪ Jb2 ∪ · · · ∪ Jbt from all Jordan chains is a basis for C n ; Pn×n = [ J1 | J2 | · · · | Jt ] is the nonsingular matrix containing these Jordan chains in the order in which they appear in J . Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 580 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 The following theorem demonstrates that the Jordan structure (the number and the size of the blocks in N ) is uniquely determined by L, but P is not. In other words, the Jordan form is unique up to the arrangement of the individual Jordan blocks. It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Uniqueness of the Jordan Structure The structure of the Jordan form for a nilpotent matrix Ln×n of index k is uniquely determined by L in the sense that whenever L is similar to a block-diagonal matrix B = diag (B1 , B2 , . . . , Bt ) in which each Bi has the form ⎛ i i .. 0 0 0 ··· 0 0 0 ⎜0 ⎜. Bi = ⎜ . ⎜. ⎝0 . ··· ··· .. . 0 0 ⎞ 0 0⎟ .⎟ .⎟ .⎟ ⎠ T H for i 0 D E ni ×ni i = 0, then it must be the case that t = dim N (L), and the number of blocks having size i × i must be given by ri−1 − 2ri + ri+1 , where ri = rank Li . IG R Proof. Suppose that L is similar to both B...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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