Information Technology Reference
In-Depth Information
21.2.3 Eigenvalues and Eigenvectors
n
×
n
, a non-zero vector
x
n
Given a square matrix
A
∈
R
∈
R
is defined as an
eigen-
vector
of the matrix if it satisfies the eigenvalue equation
=
=
Ax
λx
(x
0
)
for some scalar
λ
. In this situation, the scalar
λ
is called an
eigenvalue
of
A
corresponding to the eigenvector
x
. It is easy to verify that
αx
is also an eigenvector
of
A
for any
α
∈
R
.
To compute the eigenvalues of matrix
A
, we have to solve the following equation,
∈
R
−
=
=
(λI
A)x
0
(x
0
),
which has a non-zero solution if and only if
(λI
−
A)
is singular, i.e.,
|
λI
−
A
|=
0
.
Theoretically, by solving the above
eigenpolynomial
, we can obtain all the eigen-
values
λ
1
,λ
2
,...,λ
n
. To further compute the corresponding eigenvectors, we may
solve the following linear equation,
(λ
i
I
−
A)x
=
0
(x
=
0
).
n
×
n
, its eigenvalues
λ
1
,λ
2
,...,λ
n
and the corre-
sponding eigenvectors
x
1
,x
2
,...,x
n
, we have the following properties:
•
Given a square matrix
A
∈
R
=
i
=
1
λ
i
tr
A
=
i
=
1
λ
i
•
det
A
•
The eigenvalues of a diagonal matrix
D
=
diag
(d
1
,d
2
,...,d
n
)
are exactly the
diagonal elements
d
1
,d
2
,...,d
n
If
A
is nonsingular, then the eigenvalues of
A
−
1
•
are 1
/λ
i
(i
=
1
,
2
,...,n)
with
their corresponding eigenvectors
x
i
(i
=
1
,
2
,...,n)
.
If we denote
=
x
1
x
n
,
X
x
2
···
Λ
=
diag
(λ
1
,λ
2
,...,λ
n
),
we will have the matrix form of the eigenvalue equation
=
AX
XΛ.
If the eigenvectors
x
1
,x
2
,...,x
n
are linearly independent, then
X
is nonsingular
and we have
XΛX
−
1
.
=
A
In this situation,
A
is called
diagonalizable
.
