I'm not sure how this would fit in with your approach to singular values. My definition is that they are the diagonal entries of the matrix D in the decomposition , where U and V are unitary matrices (and the transformation T is identified with its matrix with respect to some basis). In that case, all you have to do to prove the result is to take determinants, because U and V have nonzero determinants, so , and this holds if and only if one of the diagonal elements of D is zero.