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Summary: A formal description of the matrix exponential. The definition is given as well as examples of calculating it. The matrix exponential in the frequency domain is given as well through treatment by the Laplace transform.
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Since systems are often represented in terms of matrices and solutions
of system equations often make use of the exponential, it makes sense
to try and understand how these two concepts can be combined. In many
previous applications, we've seen terms like
Recall the definition of the scalar exponential:
The definition of the matrix exponential is almost identical:
Where
Compute
And so the pattern goes, giving:
If we fill in the terms in the definition of
We notice that the sums in this matrix look familiar-in fact, they are the Taylor Series expansions of the sinusoids. Therefore, the solution further reduces to:
The example above illustrates how the use of the true definition to simplify matrix exponentials might only be easily applied in cases with inherent repetition. There is a more general method involving the Laplace Transform. In particular,
We can verify that this is true by inserting the formal definition of the matrix exponential:
The jump between the third and fourth equations here may be a
bit hard to believe, but this equality reduces to
We can do the same example as before, this time using the Laplace-based method.
Taking the inverse laplace of this gives us
In the scalar case, a product of exponentials
unless
The derivative operates on the matrix exponential the same as it does on the scalar exponential.
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This work is licensed by Thanos Antoulas and JP Slavinsky under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.
Last edited by Erin Maloney on Nov 19, 2003 8:11 pm US/Central.
