Begin2.DVI

Here Ais constant so that one can differentiate equation (10.37) with respect to t

to obtain

d

dt

(

eA t

)

=A+A^2 t+A^3

t^2

2! +A

4 t^3

3! +···+A

k tk−^1

(k−1)! +···

d

dt

(

eA t

)

=A

(

I+A t +A^2

t^2

2! +···+A

ktk

k!+···

)

d

dt

(

eA t

)

=A eA t =eA tA

(10 .38)

The exponential matrix eX is an important matrix used for solving systems of

linear differential equations. The exponential matrix has the following properties

which are stated without proofs.

1. If the matrices X and Y commute, so that XY =Y X , then one can write

eXeY =eYeX=eX+Y (10 .39)

However, if the matrices Xand Y do not commute so that XY
=Y X , then the

equation (10.39) is not true.

2.

(

eX

)− 1

=e−X

3. e[0] =I

4. eXe−X=I

5. eαX eβX =e(α+β)X

6. eX

T

=

(

eX

)T

2. The Sine Series Corresponding to the scalar sine series

sin(x t) = x t −x

(^3) t 3
3!
+x
(^5) t 5
5!
−···+ (−1)nx
2 n+1 t 2 n+1
(2 n+ 1)!
+···

there is the matrix sine series

sin(A t) = A t −A

(^3) t 3
3! +
A^5 t^5
5! +···+ (−1)
nA^2 n+1 t^2 n+1
(2 n+ 1)! +··· (10 .40)

3. The Cosine Series Corresponding to the scalar cosine series

cos(x t) = 1 −

x^2 t^2

2! +

x^4 t^4

4! −···+ (−1)

nx^2 nt^2 n

(2 n)! +···

there is the matrix cosine series

cos(A t) = I−A

(^2) t 2
2!
+A
(^4) t 4
4!
+···+ (−1)nA
2 nt 2 n
(2 n)!
+··· (10 .41)