Let A=LU and B=UL, where U is an upper triangular matrix and L is a lower triangular matrix. Demonstrate that A and B have the same eigenvalues.
The Attempt at a Solution
I know that if I can show that A and B are similar (so if I can find a matrix P such that P^(-1)AP=B) they have the same eigenvalues. But I didn't find P yet, nor do I know really how to search efficiently for P.
Another route I've thought of is to write det (I*lambda-A)=0 gives the same values for lambda as det (I*lambda-B)=0. I've thought of using det (A)=det (LU)=det L * det U = det U * det L = det B... but still can't reach anything I find useful.
Any tip is greatly appreciated.