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**Unformatted text preview: **pper-triangular matrix
possessing a simple and predictable form. This turns out to be a fundamental
problem, and the rest of this section is devoted to its solution. But before diving
in, let’s set the stage by thinking about some possibilities.
If P−1 LP = T is upper triangular, then the diagonal entries of T must
be the eigenvalues of L, so T must have the form O
C ⎛
⎜
⎜
T=⎜
⎝ 0 .. . ···
..
.
..
. ⎞
.⎟
.⎟
.
⎟.
⎠
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7.7 Nilpotent Matrices and Jordan Structure
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Please report violations to meyer@ncsu.edu One way to simplify the form of T is to allow nonzero entries only on the
superdiagonal (the diagonal immediately above the main diagonal) of T, so we
might try to construct a nonsingular P such that T has the form
⎛
⎞
0
.. ..
⎜
⎟
.
.
⎜
⎟
T=⎜
⎟.
..
⎝
⎠
.
0
To gain some insight on how this might be accomplished, let L be a 3 × 3
nilpotent matrix for which L3 = 0 and L2 = 0, and search for a P such that
⎛
⎞
⎛
⎞
010
010
P−1 LP = ⎝ 0 0 1 ⎠ ⇐⇒ L[ P∗1 P∗2 P∗3 ] = [ P∗1 P∗2 P∗3 ] ⎝ 0 0 1 ⎠
000
000
⇐⇒ LP∗1 = 0, D
E T
H LP∗2 = P∗1 , LP∗3 = P∗2 . Since L3 = 0, we can set P∗1 = L2 x for any x3×1 such that L2 x = 0. This
in turn allows us to set P∗2 = Lx and P∗3 = x. Because J = {L2 x, Lx, x}
is a linearly independent set (Exercise 5.10.8), P = [ L2 x | Lx | x ] will do the
job. J is called a Jordan chain, and it is characterized by the fact that its
ﬁrst vector is a somewhat special eigenvector for L while the other vectors are
built (or “chained”) on top of this eigenvector to form a special basis for C 3 .
There are a few more wrinkles in the development of a general theory for n × n
nilpotent matrices, but the features illustrated here illuminate the path.
For a general nilpotent matrix Ln×n = 0 of index k...

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