I'm not really interested in proofs, but I have a LTI system that is vector valued and want to take the laplace transform of it and the inverse laplace transform of the transfer function. I'm wondering how to do this with vectors? A table of vector valued Laplace transforms would be nice too if you can find. I could not. Thanks.
Edit: In particular, I am wondering more about the z-transform. Mainly you have a matrix-valued transfer function $H(z)$. How mechanically do you perform the contour integral of $H(z)$ to get $h(t)$. In particular, how to translate the z-transform tables for vectors and (although I can look this up) how to perform the partial fractions approach for vector-valued functions in the numerator and denominator. One example illustrating what to do at every step starting with a vector-valued transfer function and ending up with $h(t)$ would probably be the most helpful.