Exercise 1.45: linear difference equation model Consider the following discrete-time model x(k+ 1) = Ax(k) A__ 0.798 0.051 _ 1 ' o.715 1.088 x0" 0 (a) Compute the eigenvalues and singular values of A. See the Octave or MATLAB commands g and svd. Are the magnitudes of the eigenvalues of A less than one? Are the singular values less than one? in which (b) What is the steady state of this system? Is the steady state asymptotically stable? (c) Make a two-dimensional plot of the two components of x(k) (phase portrait) as you increase I: from k: 0 to k = 200, starting from the x(0) given above. Is 35(1) bigger than x(0)? Why or why not? . (d) When the largest eigenvalue of A is less than one but the largest singular value of A is greater than one, what happens to the evolution of x(k)? (e) Now plot the values of x for 50 points uniformly distributed on a unit circle and the corresponding Ax for these points. For the SVD corresponding to Octave and MATLAB convention , A = USV* mark 14.1, 1:2, v1, v2, 31. and 52 on your plot. Figure 1.10 gives you an idea of the appearance of the set of points for x and Ax to make sure you are on track