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**Unformatted text preview: **was sure it was on a ﬁrm mathematical foundation.
Weierstrass once said that “a mathematician who is not also something of a poet will never be
a perfect mathematician.” Copyright c 2000 SIAM Buy online from SIAM
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590
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
Jordan Form
n×n It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] For every A ∈ C
with distinct eigenvalues σ (A) = {λ1 , λ2 , . . . , λs } ,
there is a nonsingular matrix P such that
⎛ J(λ ) 0
1
⎜ 0 J(λ2 )
P−1 AP = J = ⎜ .
.
⎝.
.
.
.
0
0 ···
···
..
. ⎞ 0
0
.
.
. ⎟
⎟.
⎠ (7.8.4) D
E · · · J(λs ) • J has one Jordan segment J(λj ) for each eigenvalue λj ∈ σ (A) . • Each segment J(λj ) is made up of tj = dim N (A − λj I) Jordan
blocks J (λj ) as described below. ⎛ J (λ ) 0 · · ·
1j
⎜ 0 J2(λj ) · · ·
J(λj )=⎜ .
. ..
⎝.
.
.
.
.
0 0 T
H
⎛ ⎞ 0
0
.
.
. ⎜
⎟
⎟ with J (λj ) = ⎜
⎜
⎠
⎝ 1
..
. λj IG
R · · · Jtj(λj ) ⎞ .. . .. . ⎟
⎟
⎟.
1⎠
λj • The largest Jordan block in J(λj ) is kj × kj , where kj = index (λj ). • The number of i × i Jordan blocks in J(λj ) is given by Y
P νi (λj ) = ri−1 (λj ) − 2ri (λj ) + ri+1 (λj ) with ri (λj ) = rank (A − λj I)i .
• Matrix J in (7.8.4) is called the Jordan form for A. The structure
of this form is unique in the sense that the number of Jordan segments in J as well as the number and sizes of the Jordan blocks in
each segment is uniquely determined by the entries in A. Furthermore, every matrix similar to A has the same Jordan structure—i.e.,
A, B ∈ C n×n are similar if and only if A and B have the same
Jordan structure. The matrix P is not unique—see p. 594. O
C Example 7.8.1 ⎛ 5
⎜2
⎜
⎜0
Problem: Find the Jordan form for A = ⎜
⎜ −8
⎝
0
−8 Copyright c 2000 SIAM 4
3
−1
−8
0
−8 0
1...

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