Page 163 - Schaum's Outline of Differential Equations

146 e At [CHAP. 16

At Et
16.9. Find e e and e (A + E)t for

and

8
and verify that, for these matrices, e^e ' * e (A + B)( .
Here, A+B = Using Theorem 16.1 and the result of Problem 16.3, we find that

Thus,

16.10. Prove that e^'e ' = <? (A + B) ' if and only if the matrices A and B commute.
3
If AB = BA, and only then, we have

and, in general,

where is the binomial coefficient ("n things taken k at a time").

Now, according to Eq. (16.1), we have for any A and B:

and also

We can equate the last series in (3) to the last series in (2) if and only if (1) holds; that is, if and only if A and B
commute.

A
16.11. Prove that e^e'** = e V~ .
s)
A B
Setting t= 1 in Problem 16.10, we conclude that e e = e (A + B) if A and B commute. But the matrices At
and -As commute, since
(At)(-As) = (AA)(-ts) = (AA)(-st) = (-As)(At)
s
A
A
As
A As
Consequently, e 'g- = e< '- > = e «- >.