15. (10%) For n > 1, let Sn denote the set of all symmetric matrices and Tn denote the set of allskew-symmetric matrices, that is, Tn = {A | A is a n x n matrix such that AT = -A}. Let Ube a n x n matrix such that UTU = In. Furthermore, let
Wu ={M I M is a n x n matrix such that UTMU is a diagonal matrix}.
For a n xn matrix A, let Au denote the unique matrix B in Wu such that