Consider the following matrix. A = (i) Construct B = AA, and find the eigenvalues and eigenvectors of B. (ii) For positive eigenvalues, define of = vX;, where A1 2 12. Construct matrix D as follows. 01 D = 02 (iii) If the eigenvectors of B are not orthonormal, orthonormalise them. Make a matrix V using the orthonormal vectors you obtained. The ordering of the columns of V should be the same as the ordering of the eigenvalues, that is V = [v1 v2]. Show that V is an orthogonal matrix. (iv) Find eigenvectors of C = AAT, and orthonormalise them. (v) Find three vectors {u1, u2, us} so that W = - -Avi, i = 1,2 oi us lu1, us I u2, and |usll = 1. Compare {u1, u2, us} with the eigenvectors you found in part (iv). (vi) Orthogonalise {u1, u2, us}, and construct the matrix U = [u1 uz us], and check whether it is an orthogonal matrix. vii) Compute the product UDVT. vili) For matrix A, find a basis for C(A), C(AT), N(A), and N(AT). Compare the columns of U and V with the bases you found. What can you say about the columns of U and V in terms of the bases for the four subspaces of the matrix A? Hint: For a m x n matrix A, the column space of A is defined as C(A) = {Axx E R" } C R", and the row space of A is defined as C(AT) = {ATyly e Rm} C R". Also, the nullspace N(A) is defined as {x e R" |Ax = 0} C R". For the same matrix, the left nullspace N(A" ) is defined as {y c R"|ATy =0} C RM. Also, we have dim(C(A)) + dim(N(A")) = m dim(C(AT)) + dim(N(A)) = n