**Unformatted text preview: **lues of
A. For example, if the Jordan form for A is
⎛ O
C ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
J=⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝ 4 ⎞ 10
41
4 4
0 ⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎟
⎠ 1
4
3
0 1
3
2
2 then we know that
A9×9 has three distinct eigenvalues, namely σ (A) = {4, 3, 2};
alg mult (4) = 5, alg mult (3) = 2, and alg mult (2) = 2;
geo mult (4) = 2, geo mult (3) = 1, and geo mult (2) = 2; Copyright c 2000 SIAM Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
7.8 Jordan Form
http://www.amazon.com/exec/obidos/ASIN/0898714540 593 index (4) = 3, index (3) = 2, and index (2) = 1;
λ = 2 is a semisimple eigenvalue, so, while A is not diagonalizable, part of
it is; i.e., the restriction A/
is a diagonalizable linear operator.
N (A−2I) It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] Of course, if both P and J are known, then A can be completely reconstructed
from (7.8.4), but the point being made here is that only J is needed to reveal
the eigenstructure along with the other similarity invariants of A.
Now that the structure of the Jordan form J is known, the structure of the
similarity transformation P such that P−1 AP = J is easily revealed. Focus
on a single p × p Jordan block J (λ) contained in the Jordan segment J(λ)
associated with an eigenvalue λ, and let P = [ x1 x2 · · · xp ] be the portion of
P = [ · · · | P | · · ·] that corresponds to the position of J (λ) in J. Notice that
AP = PJ implies AP = P J (λ) or, equivalently,
⎛
⎜
⎜
A[ x1 x2 · · · xp ] = [ x1 x2 · · · xp ] ⎜
⎝ λ D
E T
H 1
..
. ⎞ . .. IG
R .. . ⎟
⎟
⎟,
1⎠ λ p×p so equating columns on both sides of this equation produces
Ax1 = λx1 =⇒ x1 is an eigenvector =⇒ (A − λI) x1 = 0, Ax2 = x1 + λx2 =⇒ (A − λI) x2 = x1 =⇒ (A − λI) x2 = 0, =⇒ (A − λI) x3 = x2
.
.
. =⇒ (A − λI) x3 = 0,
.
.
. =⇒ (A − λI) xp = xp−1 =⇒ (A − λI) xp = 0. Y
P Ax3 = x2...

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