Image Processing Reference
In-Depth Information
3.9 EIGENVALUES AND EIGENVECTORS
Spectral analysis of matrices through use of eigenvalues
eigenvectors and SVD
plays an important role in analysis and design of control systems using state-space
approach. Techniques such as state feedback by pole placement and design of state
estimators are all based on eigenvalues
=
eigenvectors decomposition. Solution of LTI
continuous and discrete systems is also directly related to functions of matrices that
is computed using eigenvalue and eigenvector decomposition.
=
3.9.1 D
EFINITION OF
E
IGENVALUE AND
E
IGENVECTOR
The nonzero vector x is an eigenvector of square n
n matrix A if there is a scale
factor
l
such that
Ax
¼ l
x
(
:
)
3
110
The scale factor
is called the eigenvalue corresponding to the eigenvector x. The
above equation can be considered as an operator operating on x. The eigenvectors
of A are vectors that are not changed by the operator, they are only scaled by
l
.
This means that eigenvectors are invariant with respect to operator A. This is
similar to the concept of eigenfunctions of LTI systems. For example the steady-
state response of an LTI system to an input of a sinusoidal signal is a sinusoidal
signal with same frequency as that of the input, but different magnitude and phase.
Therefore, sinusoidal signals are eigenfunctions of LTI systems. The equation
Ax
¼ l
lx can be written as
l
(l
I
A
)
x
¼
0
(
3
:
111
)
This equation has a nontrivial solution if and only if matrix
l
I
A is singular, that is,
det (l
I
A
) ¼
0
(
3
:
112
)
The above determinant is a polynomial of degree n and is denoted by P(l).
(l)
. This
polynomial is called the characteristic polynomial of matrix A. The characteristic
polynomial has n roots that are eigenvalues of matrix A. Corresponding to each
eigenvalue there is an eigenvector. The eigenvalues can be repeated eigenvalues and
also they may be complex.
Example 3.30
Find the eigenvalues and eigenvectors of the 2
2 matrix A:
22
A ¼
24 12
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