In Fig. 11.11 is a matrix M. It has rank 2, as you can see by observing that the first column plus the third column minus twice the second column equals 0.

(a) Compute the matrices MTM and MMT.

(b) Find the eigenvalues for your matrices of part (a).

(c) Find the eigenvectors for the matrices of part (a).

(d) Find the SVD for the original matrix M from parts (b) and (c). Note that there are only two nonzero eigenvalues, so your matrix Σ should have only two singular values, while U and V have only two columns.

(e) Set your smaller singular value to 0 and compute the one-dimensional approximation to the matrix M from Fig. 11.11.

(f) How much of the energy of the original singular values is retained by the one-dimensional approximation?