Computing Eigenvectors and Eigenvalues
1.3
2004/06/09 12:49:27 GMT-5
2004/07/30 15:18:20.103 GMT-5
Richard
G.
Baraniuk
richb@rice.edu
Richard
G.
Baraniuk
richb@rice.edu
Robert
Ahlfinger
ahlfing@rice.edu
Harika
Basana
ilsai@rice.edu
eig
matrix
An overview of the concepts behind computing eigenvectors and eigenvalues.
Step 1: Find the Eigenvalues
Find
λ
such that
A
v
λ
v
with
v
0
.
A
v
λ
v
0
A
v
λ
I
v
0
where
I
is the indentity matrix in the form
1
0
0
0
0
1
0
0
0
0
...
...
0
0
...
1
A
λ
I
v
0
This means, matrix
A
λ
I
is not invertible or
A
λ
I
0
where
˙
is the determinant.
So solve
A
λ
I
0
which will be a polynomial of
order
N
;
λ
's are its roots.
A
3
-1
-1
3
A
λ
I
3
-1
-1
3
λ
0
0
λ
3
λ
-1
-1
3
λ
A
λ
I
3
λ
-1
-1
3
λ
3
λ
2
-1
2
λ
2
6
λ
8
λ
2
λ
4
0
Therefore the eigenvalues are
λ
1
2
, and
λ
2
4
.
Step 2: Find the Eigenvectors For
each
λ
i
solve the
N
N
set of linear equations for
v
i
A
v
i
λ
v
i
with
N
equations in
N
unknowns.
A
3
-1
-1
3
,
λ
1
2
,
λ
2
4
for
λ
1
2
,
3
-1
-1
3
v
11
v
12
2
v
11
v
12
v
1
v
11
v
12
1
1
for
λ
2
4
,
3
-1
-1
3
v
21
v
22
4
v
21
v
22
v
2
v
21
v
22
1
-1
What should be done with "repeated
eigenvalues" (ie: when
λ
i
λ
j
for some
λ
i
λ
j
)?