Example 10-31. Given the matrix
A=[
2 − 1
−3 4]find the matrix function f(A) = eAt.
Solution: Here f(x) = ext,and from the previous example, the eigenvalues of Ahave
been determined as λ 1 = 1 and λ 2 = 5.If the matrix equation is to be represented in
the form
f(A) = R(A) = γ 1 A+γ 2 Iwhich is a linear combination of {I, A }then one can use the equation (10.46) and
write
f(λ 1 ) = eλ^1 t=et=γ 1 (1) + γ 2
f(λ 2 ) = eλ^2 t=e^5 t=γ 1 (5) + γ 2(10 .48)From these equations it is possible to solve for γ 1 and γ 2 and show
γ 1 =e5 t−et4 , and γ^2 =
5 et−e^5 t
4.The matrix function for f(A)can then be represented as
f(A) = eAt =(
e^5 t−et
4)
A+(
5 et−e^5 t
4)
Iwhich has the equivalent matrix form
eAt =^1
4[
(e^5 t+ 3et) (et−e^5 t)
(3 et− 3 e^5 t) (3 e^5 t+et)]
A=[
2 − 1
−3 4]Example 10-32. Find the matrix function
f(A) = sin At for A=
0 1 0
1 0 1
1 1 1
Solution: The characteristic equation of Ais
C(λ) = |A−λI |=∣∣
∣∣
∣∣−λ 1 0
1 −λ 1
1 1 1 −λ∣∣
∣∣
∣∣= (λ(^2) −1)(1 −λ) = 0.
The eigenvalues of Aare
λ 1 = 1, λ 2 = 1, λ 3 =− 1.