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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
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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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