Differentiate equation (10.40) with respect to tand show
d
dt sin(At) = A−A3 t^2
2! +A5 t^4
4! +···+ (−1)nA 2 n+1 t^2 n
(2 n)! +···
d
dt sin(At) = A(
I−A^2 t^2
2! +A^4 t^4
4! +···+ (−1)nA^2 nt^2 n
(2 n)! +···)d
dt sin(At) = Acos(A t) = cos(A t)A(10 .42)Differentiate equation (10.41) with respect to tand show
d
dtcos(A t) = −A^2 t+A^4 t3
3!+···+ (−1)nA^2 n t2 n− 1
(2 n−1)!+···d
dtcos(A t) = −A(
A t −A(^3) t 3
3!
+A
(^5) t 5
5!
+···+ (−1)nA
2 n+1 t 2 n+1
(2 n+ 1)!
+···
)
d
dt
cos(A t) = −Asin(A t) = −sin(A t)A
(10 .43)
Example 10-28. Show that if Ais a constant matrix, then X(t) = eA(t−t^0 ) is a
matrix solution to the initial-value problem to solve
dX
dt=AX, X (t 0 ) = ISolution Differentiate X and show dX
dt=Ae A(t−t^0 ) =AX is satisfied. At the initial
time t=t 0 , one finds X(t 0 ) = eA(0) =I.
The Hamilton-Cayley Theorem
The Hamilton^4 -Cayley^5 theorem states that every n×nconstant square matrix
Asatisfies its own characteristic equation. That is, if C(λ) = 0 is the characteristic
equation association with a n×nsquare matrix A, then the equation C(A) = [0] must
be satisfied. This result is known as the Hamilton-Cayley theorem.
(^4) William Rowan Hamilton (1806–1865),Irish mathematician and physicist.
(^5) Arthur Cayley (1821–1895),English mathematician.