579 that there is a nonsingular matrix y1 such that n

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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). i=1 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 581 It is illegal to print, duplicate, or distribute this material 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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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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