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

The Matrix Exponential via the Laplace Transform

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

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

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

eat=-11sa a t 1 s a
(1)
The natural matrix definition is therefore
eAt=-1sIA-1 A t 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=( 10 02 ) A 1 0 0 2 for then sIA-1=( 1s10 01s2 ) s I A 1 s 1 0 0 1 s 2 and so (recalling Equation 1 above) eAt=( -11s10 0-11s2 )=( et0 0e2t ) A t 1 s 1 0 0 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=( 01 00 ) 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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Definition of a lens

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