Simplifying e^{At} to Matrix Form: A General Expression?

[epkid08]

I did a problem in class today that evaluated $$f(t)=e^{At}$$ for $$A_{2,2}=\begin{bmatrix}2&1 \\-1&4 \end{bmatrix}$$ to a matrix form.

The answer I got was:

$$f(t)=\begin{bmatrix}e^{3t}-te^{3t}&te^{3t} \\-te^{3t}&e^{3t}+te^{3t} \end{bmatrix}$$

Factoring we have:

$$f(t)=e^{3t}\begin{bmatrix}1-t&t \\-t&1+t \end{bmatrix}$$

My question is, is there some simple general expression for simplifying $$e^{At}$$ to a matrix form? Maybe something that resembles $$e^{tA_{2,2}}=e^{\lambda t}\begin{bmatrix}1-t&t \\-t&1+t \end{bmatrix}$$

but for any size matrix.

 

[tiny-tim]

Hi epkid08!

If you can write A in the form PQP^-1 where Q is diagonal …

then ∑ Aⁿ/n! = P(∑ Qⁿ/n!)P^-1 = Pe^QP^-1, where e^Q = … ?

(oh, and your simple form with a single exponential factor on the outside only works in thsi case because there is a double eigenvalue )