Page 426 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

Thus, A = 0 is the eigenvalues of A with multiplicity 3. By Eq. (7.66) we have
eA' = b,I + b,A + b2A2

where b,, b,, and b2 are determined by setting A = 0 in the following equations [App. A, Eqs.
(A. 59) and (A. 60 )I:

b, + b,A + b2h2 = eA'

Thus,

Hence,

t
eA' = I + tA + -A'
2
which is the same result obtained in Prob. 7.44(b).

7.46. Show that

eA+B = eAeB
provided A and B commute, that is, AB = BA.
By Eq. (7.53)

1 1 1 1
=I+A+B+-A~+-AB+-BA+-B~+ ...
2! 2 2 2!
and eAeB - eA+B = L , (AB - BA) + .
Thus, if AB = BA, then
eA+B = eAeB

7.47. Consider the matrix