Civil Engineering Reference
In-Depth Information
aaa
a
x
x
x
0
0
0
0
0
11
12
13
1
n
1
aaa
a
21
22
23
2
n
2
aaa
a
=
31
32
33
3
n
3
aaa
a
x
n
n
1
n
2
n
3
nn
Let us consider the solution of eigenvalue problems. For any square
matrix, [
A
], the determinant equation |
A
−
l
I
|=0 is a polynomial equation of
degree
n
unknowns in the variable
l
. In other words, there are exactly
n
roots that satisfy this equation. These roots are known as
eigenvalues
of
A
.
(
)
+
…
+
a
−
l
x ax ax
+
+
ax
=
0
11
1
122 13
3
1
nn
(
)
+
…
+
ax a
+−
l
x ax
+
a
x
=
0
0
21 1
22
2
23
3
2
n
n
(
)
+
…
+
ax ax a
+
+
−
l
x
ax
=
31 1
322 33
3
3
nn
ax ax ax
+
…
+−
(
)
+
+
a
l
x
=
0
n
11 22 33
n
n
nn
n
Converting these equations into matrix form, [
A
−
l
I
][
x
]=0:
(
)
a
−
l
a
a
a
x
x
x
0
0
0
0
0
11
12
13
1
n
1
(
)
a
a
−
l
a
a
21
22
23
2
n
2
(
)
a
a
a
−
l
a
=
31
32
33
3
n
3
(
)
a
a
a
a
−
l
x
n
1
n
2
n
3
nn
n
The non-trivial solution exists if the determinant of the coefficient
matrix is zero. We use this so that Cramer's rule can be used to find the
eigenvalues. Example 2.15 shows the process of determining the eigen-
values by Cramer's rule.
Example 2.15
Eigenvalues by Cramer's rule
Determine the eigenvalues for the following set of equations using
Cramer's rule.
0230
10 120
2470
++=
−−+=
−++=
x
x x
xx x
1
2
3
1
2
3
x
x
x
1
2
3
