The desired similarity transformation is pnn j1 j2

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Unformatted text preview: ∩ N (L) ⊆ R Li . A Jordan chain is built on top of each b ∈ Si by solving the system Li x = b for x and by setting IG R Jb = {Li x, Li−1 x, . . . , Lx, x}. Y P (7.7.2) Notice that chains built on top of vectors from Si each have length i + 1. The heuristic diagram in Figure 7.7.2 depicts Jordan chains built on top of the basis vectors illustrated in Figure 7.7.1—the chain that is labeled is built on top of a vector b ∈ S3 . O C Figure 7.7.2 Copyright c 2000 SIAM 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 577 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu The collection of vectors in all of these Jordan chains is a basis for C n . To demonstrate this, first it must be argued that the total number of vectors in all Jordan chains is n, and then it must be proven that this collection is a linearly independent set. To count the number of vectors in all Jordan chains Jb , first recall from (4.5.1) that the rank of a product is given by the formula rank (AB) = rank (B) − dim N (A) ∩ R (B), and apply this to conclude that dim Mi = dim R Li ∩ N (L) = rank Li − rank LLi . In other words, if we set di = dim Mi and ri = rank Li , then di = dim Mi = rank Li − rank Li+1 = ri − ri+1 , (7.7.3) D E so the number of vectors in Si is νi = di − di+1 = ri − 2ri+1 + ri+2 . (7.7.4) Since every chain emanating from a vector in Si contains i + 1 vectors, and since dk = 0 = rk , the total number of vectors in all Jordan chains is k−1 total = T H k−1 (i + 1)(di − di+1 ) (i + 1)νi = i=0 i=0 IG R = d0 − d1 + 2(d1 − d2 ) + 3(d2 − d3 ) + · · · + k (dk−1 − dk ) = d0 + d1 + · · · + dk−1 = (r0 − r1 ) + (r1 − r2 ) + (r2 − r3 ) + · · · + (rk−1 − rk ) = r0 = n. Y P To prove that the set of all vectors from all Jordan chains is linearly independent, place the...
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