**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
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7.7 Nilpotent Matrices and Jordan Structure
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Please report violations to [email protected] 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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