Determinant of the matrix exponential

[syj]

Homework Statement

Show that det(e^A)=e^tr(A) for A$\in$C_nxn

Homework Equations

The Attempt at a Solution

I am sooo bad at proofs.
And I am still trying to wrap my brain around the concept of matrix exponentials.
Can someone please get me started ...

 

[micromass]

Remember that the determinant is the product of the eigenvalues and that the trace is the sum of the eigenvalues...

Also, you should work with the Jordan canonical form of A.

 

[syj]

Ok
So let A=PDP^-1 where D is the Jordan Canonical form of A
then e^A=Pe^DP^-1

Now where to from here??

 

[HallsofIvy]

If A is in Jordan Normal Form, what does $e^A$ look like? What is $det(e^A)$?

You should be able to see that, since the Jordan Normal Form of any matrix is an "upper triangular matrix", all powers of the Jordan Normal Form is also an upper triangular matrix and so is the exponential. Think about how you would find the determinant of any upper triangular matrix.