The Matrix Exponential via The Laplace Transform 2.9 2002/06/21 2002/07/10 Steven Cox cox@rice.edu Liqun Wang liqun@rice.edu Steven Cox cox@rice.edu exponential Laplace transform matrix This module introduces how to use Laplace transform to calculte the matrix exponential. You may recall from the Laplace Transform module that may achieve a t via a t ℒ 1 s a The natural matrix definition is therefore A t ℒ s I A where I is the n-by-n identity matrix. The easiest case is the diagonal case, e.g., A 1 0 0 2 for then s I A 1 s 1 0 0 1 s 2 and so (recalling above) A t ℒ 1 s 1 0 0 ℒ 1 s 2 t 0 0 0 2 t As a second example let us suppose 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)] ]]> If 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] ]]>