**Unformatted text preview: **and N, where B is as described
above and N is as described in (7.7.6). This implies that B and N are similar,
and hence rank Bi = rank Li = ri for every nonnegative integer i. In
particular, index (B) = index (L). Each time a block Bi is powered, the line of
i ’s moves to the next higher diagonal level so that Y
P O
C rank (Bp ) =
i Since rp = rank (Bp ) =
i × i blocks in B, then t
i=1 ni − p
0 if p < ni ,
if p ≥ ni . rank (Bp ), it follows that if ωi is the number of
i rk−1 = ωk ,
rk−2 = ωk−1 + 2ωk ,
rk−3 = ωk−2 + 2ωk−1 + 3ωk ,
.
.
. and, in general, ri = ωi+1 + 2ωi+2 + · · · + (k − i)ωk . It’s now straightforward
to verify that ri−1 − 2ri + ri+1 = ωi . Finally, using this equation together with
(7.7.4) guarantees that the number of blocks in B must be
k t=
i=1 Copyright c 2000 SIAM k k (ri−1 − 2ri + ri+1 ) = ωi =
i=1 νi = dim N (L).
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7.7 Nilpotent Matrices and Jordan Structure
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Please report violations to [email protected] The manner in which we developed the Jordan theory spawned 1’s on the superdiagonals of the Jordan blocks Ni in (7.7.5). But it was not necessary to
do so—it was simply a matter of convenience. In fact, any nonzero value can be
forced onto the superdiagonal of any Ni —see Exercise 7.7.9. In other words,
the fact that 1’s appear on the superdiagonals of the Ni ’s is artiﬁcial and is not
important to the structure of the Jordan form for L. What’s important, and
what constitutes the “Jordan structure,” is the number and sizes of the Jordan
blocks (or chains) and not the values appearing on the superdiagonals of these
blocks. Example 7.7.1 D
E Problem: Determine the Jordan forms for 3 × 3 nilpotent matrices L1 , L2 ,
and L3 that have respective indices k = 1, 2, 3.
Solution: The size of the largest block must be k × k, so
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