Page 427 - Schaum's Outline of Theory and Problems of Signals and Systems

STATE SPACE ANALYSIS [CHAP. 7

Now we decompose A as

A=A+N
0 1 0
[200]
where A=O2 0 and [ N=OO 1
0 0 2 0 0 0

(a) Show that the matrix N is nilpotent of index 3.
(b) Show that A and N commute, that is, AN = Nh
(c) Using the results from parts (a) and (b), find eA'.

(a) By direct multiplication we have
0 1 0 0 1 0 0 0
N~=NN= o o 1 o o 1 = o o
lo 0 olio 0 01 (0 0

0 0 1 0 1 0 0 0
N3=N2N=0 0 0 0 0 1 =0 0
[o 0 oI[o 0 01 [o 0
Thus, N is nilpotent of index 3.
(b) Since the diagonal matrix A can be expressed as 21, we have

that is, A and N commute.
(c) Since A and N commute, then, by the result from Prob. 7.46

Now [see App. A, Eq. (A.4911

and using similar justification as in Prob. 7.44(b), we have

Thus,