Graphics Programs Reference
In-Depth Information
Symmetric Matrix Eigenvalue Problems
9
Find
λ
forwhich nontrivial solutionsof
Ax
=
λ
x
exist
9.1
Introduction
The
standard form
of the matrix eigenvalue problemis
Ax
=
λ
x
(9.1)
where
A
is a given
n
×
n
matrix. The problemistofind the scalar
λ
and the vector
x
.
Rewriting Eq. (9.1) in the form
(
A
−
λ
I
)
x
=
0
(9.2)
it becomes apparentthat we are dealing with a system of
n
homogeneousequations.
An obvious solutionis the trivialone
x
0
.Anontrivial solution can exist onlyif the
determinant of the coefficient matrix vanishes; that is, if
=
|
A
−
λ
I
| =
0
(9.3)
Expansion of the determinantleads to the polynomialequation known as the
characteristic equation
n
n
−
1
a
1
λ
+
a
2
λ
+···+
a
n
λ
+
a
n
+
1
=
0
which has the roots
λ
i
,
i
=
1
,
2
,...,
n
,called the
eigenvalues
of the matrix
A
. The
−
λ
i
I
)
x
=
solutions
x
i
of (
A
0
areknown as the
eigenvectors
.
As an example, consider the matrix
⎡
⎣
⎤
⎦
1
−
1
0
A
=
−
12
−
1
(a)
0
−
11
326
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