Proving diagonalizability with the power series

[rugbygirl2]

Homework Statement

If A is a diagonal matrix with the diagonal entries a1, a2, ..., an, use the power series to prove that exp(At) is a diagonal matrix with the entries exp(a1t), exp(a2t), ..., exp(ant).

Homework Equations

The Attempt at a Solution

I can prove that A is diagonalizable with T-1 AT=D, but I'm not sure how to begin the proof using the power series as a method. My book does not discuss this and any insight would be greatly appreciated, thanks!

 

[Dick]

You don't have to diagonalize A. It's already diagonal. You just have to realize that the power series of A is the same as the power series of the diagonal elements.