Is the trace of a matrix preserved by an orthogonal transformation?

[ehrenfest]

Homework Statement

My statistical mechanics book says that if M is an real, symmetric n by n matrix, and U is the matrix of its eigenvectors as column vectors, then the transformation UMU^{-1} preserves the trace of M. Is that true? If so, is it obvious? If it is true but not obvious, how do you prove it?

Homework Equations

The Attempt at a Solution

 

[Dick]

It's true and obvious. And U doesn't need to be unitary. Use the cyclic property of trace. Tr(ABC)=Tr(CAB).

 

[ehrenfest]

Tr(ABC)=Tr(CAB).

Why is that true?

EDIT: never mind http://en.wikipedia.org/wiki/Trace_(linear_algebra )