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Homework Problems

The function of a matrix can formally be defined through a Taylor series
expansion. For example, the exponential of a matrix M can be defined through:

∞
exp(M = ∑ M n
)
n =0 ! n
Pb. 8.17 Use the results from Pb. 8.15 to show that:

exp(M) = V exp(D)V –1

where, for any diagonal matrix:

 λ 1 0 L L 0   exp(λ 1 ) 0 L L 0 
 0 λ M   0 exp(λ ) M 
 2   2 

exp M O M  =  M O M 
 M λ   M exp(λ 
0
0
   0 0 L n− 1 λ       0 0 L 0 n− 1 ) exp(λ   
0
)
n
n
Pb. 8.18 Using the results from Pb. 8.17, we deduce a direct technique for
solving the initial value problem for any system of coupled linear ODEs with
constant coefficients.
Find and plot the solutions in the interval 0 ≤ t ≤ 1 for the following set of
ODEs:
dx
1 = x + 2 x
dt 1 2
dx
2 = 2 x − 2 x
dt 1 2

with the initial conditions: x (0) = 1 and x (0) = 3. (Hint: The solution of
1 2
dX
= AX is X( t =) exp( A t X) ( ),0 where X is a time-dependent vector and A is
dt
a time-independent matrix.)
Pb. 8.19 MATLAB has a shortcut for computing the exponential of a matrix.
While the command exp(M) takes the exponential of each element of the
matrix, the command expm(M) computes the matrix exponential. Verify
your results for Pb. 8.18 using this built-in function.

© 2001 by CRC Press LLC