A geometrically inspired proof of the singular value decomposition

Any real n X m matrix A can be factored into A = USV' where U and V are n X n and m x m real orthogonal matrices and S is a diagonal n X m matrix for which sl > S22> ***> Spp >0 and p = min(m, n). This is the singular value decomposition (SVD) of A and the si are its singular values. An equivalent formulation of the SVD for linear transformations can be given as follows. For any linear transformation A: Rm -R n there exist orthonormal bases Vl, V2... vm in R' and ul, u2,..., un in Rn such that Avi = siiuj for i = 1,..., p and Av1 = 0 for i = p + 1,..., m. This can be viewed geometrically by saying that A takes the coordinate system of v's to the coordinate system of u's with expansions and contractions along corresponding coordinate directions given by the