In an exercise I am asked to find the eigenvalues of a matrix A by demanding that a unitary matrix (see the attached file) diagonalizes it. I know I could just solve the eigenvalue equation but I think I am supposed to do it this rather tedious way. Now I have introduced an arbitrary unitary matrix and performed the matrix multiplication: U^(dagger)AU = D, where D is a diagonalized matrix. From this last relation I should somehow be able to find the eigenvalues, but I haven't been able to so far. Have I done it wrong or is there actually a way through this messy algebra. What should I do? Note that the unitarity demands that lu_kl^2+lv_kl^2 = 1, which I suspect could be useful, but are there any other relations, which I am missing?