Laplace transform of a matrix exponential

[A_B]

Homework Statement

show that the Laplace transform of e^(At) = (sI - A)^(-1)

$$\mathcal{L}\left\{ e^{At} \right\}(s) = \left(sI - A \right)^{-1}$$

The Attempt at a Solution

I find
$$\left( e^{At} \right)_{ij} = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}t^k}{k!}$$

and since
$$\mathcal{L}\left\{ (A^k)_{ij}t^k \right\}(s) = \frac{k!}{s^{k+1}} (A^k)_{ij}$$

we have
$$\mathcal{L}\left\{\left( e^{At} \right)_{ij}\right\}(s) = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}}{s^{k+1}}$$

and there I'm stuck.

Thanks
A_B

 

[Dick]

First show it's true for a diagonal matrix D. Then show it's still true if A is diagonalizable, i.e. PAP^(-1)=D for some invertible matrix P.

 

[Ray Vickson]

Homework Statement

show that the Laplace transform of e^(At) = (sI - A)^(-1)

$$\mathcal{L}\left\{ e^{At} \right\}(s) = \left(sI - A)^{-1} \right)$$

The Attempt at a Solution

I find
$$\left( e^{At} \right)_{ij} = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}t^k}{k!}$$

and since
$$\mathcal{L}\left\{ (A^k)_{ij}t^k \right\}(s) = \frac{k!}{s^{k+1}} (A^k)_{ij}$$

we have
$$\mathcal{L}\left\{\left( e^{At} \right)_{ij}\right\}(s) = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}}{s^{k+1}}$$

and there I'm stuck.

Thanks
A_B

$$(sI-A)^{-1}= \frac{1}{s}(I-\frac{1}{s}A)^{-1}.$$

RGV