**Unformatted text preview: **⎞
⎛
z0
⎜ z2 ⎟
= [ 0 | 0 | · · · | Lk−1 Qk−1 ] ⎜ . ⎟ =⇒ zk−1 ∈ N Lk−1 Qk−1
⎝.⎠
. O
C Y
P =⇒ zk−1 zk−1 = 0. This conclusion with the same argument applied to 0 = Lk−2 Qz produces
zk−2 = 0. Similar repetitions show that zi = 0 for each i, and thus N (Q) = 0.
It has now been proven that if B = Sk−1 ∪ Sk−2 ∪ · · · ∪ S0 = {b1 , b2 , . . . , bt }
is the basis for N (L) derived from the nested subspaces Mi , then the set of
all Jordan chains J = Jb1 ∪ Jb2 ∪ · · · ∪ Jbt is a basis for C n . If the vectors
from J are placed as columns (in the order in which they appear in J ) in a
matrix Pn×n = [ J1 | J2 | · · · | Jt ], then P is nonsingular, and if bj ∈ Si , then
Jj = [ Li x | Li−1 x | · · · | Lx | x ] for some x such that Li x = bj so that
⎛
⎞
01
.. ..
⎜
⎟
.
.
⎜
⎟
LJj = [ 0 | Li x | · · · | Lx ] = [ Li x | · · · | Lx | x ] ⎜
. . 1 ⎟ = Jj Nj ,
⎝
⎠
.
0 Copyright c 2000 SIAM Buy online from SIAM
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7.7 Nilpotent Matrices and Jordan Structure
http://www.amazon.com/exec/obidos/ASIN/0898714540 579 where Nj is an (i + 1) × (i + 1) matrix whose entries are equal to
superdiagonal and zero elsewhere. Therefore,
⎛
N1 0 · · ·
⎜ 0 N2 · · ·
LP = [ LJ1 | LJ2 | · · · | LJt ] = [ J1 | J2 | · · · | Jt ] ⎜ .
..
⎝.
.
.
0 1 along the
0
0
.
.
. ⎞
⎟
⎟
⎠ · · · Nt 0 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] or, equivalently,
⎛ N1 0 · · ·
⎜ 0 N2 · · ·
P−1 LP = N = ⎜ .
..
⎝.
.
.
0 0 ··· ⎛
⎞
0
0
⎜
0⎟
⎜
. ⎟ , where Nj = ⎜
.⎠
⎝
.
Nt 1
..
. ⎞ .. . .. . D
E ⎟
⎟
⎟ . (7.7.5)
1⎠
0 T
H Each Nj is a nilpotent matrix whose index is given by its size. The Nj ’s are
called nilpotent Jordan blocks, and the block-diagonal matrix N is called the
Jordan form for L. Below is a summary. IG
R Jordan Form for a Nilpotent Matrix Every nilpotent matrix Ln×n of index k is similar to a bloc...

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