Laplace transform table for matrix?

Question

Given a homogeneous linear time invariant dynamical system x' = Ax, by performing Laplace transform,

$L[x'(t)] = L[Ax(t)]$

$sX - x(0) = AX$

$sX - AX = x(0)$

$(sI - A)X = x(0)$

$X = (sI-A)^{-1}x(0)$

and by after taking a inverse Laplace transform,

$x(t) = e^{At}x(0)$ where $e^{At}$ is a matrix exponential,

that $(sI-A)^{-1}$ is in the similar form as Laplace transform in scalar case $\frac{1}{s-a}$, whose inverse Laplace transform is $e^{at}$.

This seems to suggest that other transforms from the scalar Laplace transform table are also applicable to matrix cases. Is this idea true?

Answer 1

You should look into functional calculus: large classes of identities among functions of a single scalar variable can be systematically generalized to functions of a single matrix variable.