Mathematical Methods for Physics and Engineering - Matematica.NET

8.18 SIMULTANEOUS LINEAR EQUATIONS◮Show that, for i =1, 2,...,p, Av i = s i u i and A † u i = s i v i ,wherep =min(M,N).Post-multiplying both sides of (8.131) by V, and using the fact that V is unitary, we obtainAV = US.Since the columns of V and U consist of the vectors v i and u j respectively and S has onlydiagonal non-zero elements, we find immediately that, for i =1, 2,...,p,Av i = s i u i . (8.134)Moreover, we note that Av i =0fori = p +1,p+2,...,N.Taking the Hermitian conjugate of both sides of (8.131) and post-multiplying by U, weobtainA † U = VS † = VS T ,where we have used the fact that U is unitary and S is real. We then see immediately that,for i =1, 2,...,p,A † u i = s i v i . (8.135)We also note that A † u i =0fori = p +1,p+2,...,M. Results (8.134) and (8.135) are usefulfor investigating the properties of the SVD. ◭The decomposition (8.131) has some advantageous features for the analysis ofsets of simultaneous linear equations. These are best illustrated by writing thedecomposition (8.131) in terms of the vectors u i and v i asp∑A = s i u i (v i ) † ,i=1where p = min(M,N). It may be, however, that some of the singular values s iare zero, as a result of degeneracies in the set of M linear equations Ax = b.Let us suppose that there are r non-zero singular values. Since our convention isto arrange the singular values in order of decreasing size, the non-zero singularvalues are s i , i =1, 2,...,r, and the zero singular values are s r+1 ,s r+2 ,...,s p .Therefore we can write A asr∑A = s i u i (v i ) † . (8.136)i=1Let us consider the action of (8.136) on an arbitrary vector x. This is given byr∑Ax = s i u i (v i ) † x.i=1Since (v i ) † x is just a number, we see immediately that the vectors u i , i =1, 2,...,r,must span the range of the matrix A; moreover, these vectors form an orthonormalbasis for the range. Further, since this subspace is r-dimensional, we haverank A = r, i.e. the rank of A is equal to the number of non-zero singular values.The SVD is also useful in characterising the null space of A. From (8.119),we already know that the null space must have dimension N − r; so,ifA has r303