Relation between commutator, unitary matrix, and hermitian exponential operator

[silverdiesel]

Homework Statement

Show that one can write U=exp(iC), where U is a unitary matrix, and C is a hermitian operator. If U=A+iB, show that A and B commute. Express these matrices in terms of C. Assum exp(M) = 1+M+M^2/2!...

Homework Equations

U=exp(iC)
C=C*
U*U=I
U=A+iB
exp(M) = sum over n: ((M)^n)/n!

The Attempt at a Solution

I am really stumped. I tried (A+iB)(A*-iB*)=I, and I can get the commutator to come out of that, but I have these A*A and B*B terms which I am unsure how to use. I am also not using the exponential term in any way. I know it has something to do with the taylor expansion, just not sure how to get A+iB into that expansion.

 

[Latrace]

Remember that A and B are real matrices. For the first question, try diagonalizing U first.