Civil Engineering Reference
In-Depth Information
(
)
0
−
l
2
3
(
)
−
10
− −
1
l
2
=
0
(
)
−
2
4
7
−
l
Solve the determinant by the basket weave method.
(
)
(
)
0
−
l
2
3
0
−
l
2
(
)
(
)
−
10
− −
1
l
2
−
10
− −
1
l
(
)
−
2
4
7
−
l
−
2
4
(
)
−−
(
)
(
)
+
()()
−
()
+
()
−
(
)()
0
−
l
1
l
7
−
l
2 22 3 04
−
()
−−
(
)
−
()
(
)()()
−
()
−
(
)
(
)
31 2
l
−−
0 24 2 07
l
−
l
l
3
2
6160
−+−=
l
l
The solution to the cubic equation can be found by many of the methods
from Chapter 1 and represent the eigenvalues
l
= 1, 2, and 3.
2.14
fADDEEV-LEVERRiER MEtHOD
The Faddeev-Leverrier method is a polynomial method used to find the
eigenvalues. The method is named for Dmitrii Konstantinovich Faddeev
and published by Urbain Jean Joseph Le Verrier in 1840 (Le Verrier
1839). From linear algebra, the trace of a matrix is the sum of the diag-
onal terms. The process for determining the characteristic polynomial is
as follows:
(
)
=
()
−
n
−
…
−
n
n
−
1
n
−
2
n
−
3
1
l
p
l
−
p
l
−
p
l
p
0
1
2
3
n
[]
=
[]
=
[]
BA
ptrB
1
1
1
1
(
)
[]
=
[]
[]
−
[]
2
=
[]
BABpI
p
trB
2
1
1
2
2
1
(
)
[]
=
[]
[
−
[]
3
=
[]
BABpI
p
trB
3
2
2
3
3
1
(
)
[]
=
[]
[
]
−
[]
=
[]
BABp
I
p
k
tr B
k
k
−
1
k
−
1
k
k
1
(
)
[]
=
[]
n
[
]
−
[]
=
[]
BAB
pI
p
n
tr B
−
1
n
−
1
n
n
