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The Matrix Exponential via the Laplace Transform

Module by: Steven J. Cox. E-mail the author

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Summary: This module introduces how to use Laplace transform to calculte the matrix exponential.

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You may recall from the Laplace Transform module that may achieve ⅇat a t via

ⅇat=ℒ-11s−a a t ℒ 1 s a (1)
The natural matrix definition is therefore
ⅇAt=ℒ-1sI−A-1 A t ℒ s I A (2)
where II is the n-by-n identity matrix.

Example 1

The easiest case is the diagonal case, e.g., A=1002 A 1 0 0 2 for then sI−A-1=1s−1001s−2 s I A 1 s 1 0 0 1 s 2 and so (recalling Equation 1 above) ⅇAt=ℒ-11s−100ℒ-11s−2=ⅇt00ⅇ2t A t ℒ 1 s 1 0 0 ℒ 1 s 2 t 0 0 0 2 t

Example 2

As a second example let us suppose A=01-10 A 0 1 -1 0 and compute, in matlab,


	
	
	>> inv(s*eye(2)-A)  
	
	   ans = [ s/(s^2+1),  1/(s^2+1)]
	         [-1/(s^2+1),  s/(s^2+1)]

	>> ilaplace(ans)

	   ans = [ cos(t),  sin(t)]
	         [-sin(t),  cos(t)]

Example 3

If A=0100 A 0 1 0 0 then 


	
	
	>> inv(s*eye(2)-A)  
	
	   ans = [ 1/s,  1/s^2]
	         [   0,    1/s]

	>> ilaplace(ans)

	   ans = [ 1,  t]
	         [ 0,  1]

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