Section 9–5 / Some Useful Results in Vector-Matrix Analysis 671Noting that
we find
Computation of eAt: Method 2. The second method of computing uses the
Laplace transform approach. Referring to Equation (9–36), can be given as follows:
Thus, to obtain first invert the matrix This results in a matrix whose
elements are rational functions of s.Then take the inverse Laplace transform of each
element of the matrix.
EXAMPLE 9–7 Consider the following matrix A:
Compute by use of the two analytical methods presented previously.Method 1. The eigenvalues of Aare 0 and –2 A necessary transformation
matrixPmay be obtained asThen, from Equation (9–46), is obtained as follows:Method 2. Sincewe obtain(s I-A)-^1 =D
1
s01
s(s+2)
1
s+ 2T
s I-A=B
s
00
sR - B
0
0
1
- 2
R =B
s
0- 1
s+ 2R
eAt= B
1
0
1
- 2
RB
e^0
00
e-2tRB
1
0
1
2-^12
R= B
1
0
1
2 A^1 - e- 2tB
e-2tR
eAtP= B
1
0
1
- 2
R
Al 1 =0, l 2 =- 2 B.eAtA= B
0
0
1
- 2
R
eAt, (s I-A).
eAt=l-^1 C(s I-A)-^1 D
eAt
eAt
=C
et-tet+^12 t^2 et
12 t
(^2) et
tet+^12 t^2 et
tet-t^2 et
et-tet-t^2 et
- 3tet-t^2 et
12 t
(^2) et
tet+^12 t^2 et
et+2tet+^12 t^2 et
S
=C
1
1
1
0
1
2
0
0
1
SC
et
0
0
tet
et
0
12 t
(^2) et
tet
et
SC
1
- 1
1
0
1
- 2
0
0
1
S
eAt=SeJt S-^1
eJt= C
et
0
0
tet
et
0
12 t
(^2) et
tet
et
S