Implement the proof of Theorem 2.7 by completing MATLAB Algorithm 2.4, which takes as input a rectangular matrix A and outputs two orthogonal matrices U 2 Rmm, and V 2 Rnn and a diagonal matrix S 2 Rmn consisting of the singular values of A, such that AV = US. The procedure uses MATLAB schur(X,real) and the already written Algorithm 2.2. 2. Test Algorithm 2.4 by generating a rectangular matrix using the function and compare the results with those obtained using MATLAB svd.

Algorithm 2.4 Singular Value Decomposition Using MATLAB Commands

function [U S V] = mysvd(A)

[m n] = size(A);

[U T] = schur(A*A,real);

%AAU=U*T S = zeros(m,n);

r=rank(T);

% r<=m s=sqrt(diag(T));

s=sort(s,descend);

%%%%%%%%%%%%%% %%%% COMPLETE THIS PART OF THE PROGRAM BY DOING AN ECONOMY SVD THAT GIVES Ur, Vr, Sr, THEN COMPLETE Vr using a modification of the Classical Gram-Schmidt (Ref. Computer Exercise 2.1) %%%%%%%%%%%%%% %%%%

Theorem 2.7 Let A 2 Cnn. There exists an upper triangular matrix T 2 Cnn and a unitary matrix U 2 Cnn (UU = UU = I), such that A = UTU i.e., AU = UT, (2.7) i.e., the triangular matrix T is similar2 to A.