Solved: Set S = round(10 rand(5)); S = triu(S, 1) + eye(5) S = S S T = inv(S) (a) The

Chapter 6, Problem 19

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Set S = round(10 rand(5)); S = triu(S, 1) + eye(5) S = S S T = inv(S) (a) The exact inverse of S should have integer entries. Why? Explain. Check the entries of T using format long. Round the entries of T to the nearest integer by setting T = round(T). Compute T S and compare with eye(5). (b) Set A = triu(ones(5), 1) + diag(1 : 5), B = S A T The matrices A and B both have the eigenvalues 1, 2, 3, 4, and 5. Use MATLAB to compute the eigenvalues of B. How many digits of accuracy are there in the computed eigenvalues? Use MATLAB to compute and compare each of the following: (i) det(A) and det(B) (ii) trace(A) and trace(B) (iii) SA2T and B2 (iv) SA1T and B1 results, using format long. Are the computedeigenvalues real? Compute also the eigenvectorsby setting[X, D] = eig(A)What type of matrix would you expect X tobe? Use the MATLAB command X X tocompute XHX. Do the results agree with yourexpectations?(b) SetE = D+j eye(5) and B = X E/XWhat type of matrix would you expect B to be?Use MATLAB to compute BHB and BBH. Howdo these two matrices compare?Visualizing the Singular Value DecompositionIn some of the earlier exercises we used MATLABseigshow command to look at geometric interpretationsof the eigenvalues and eigenvectors of 2 2matrices. The eigshow facility also has an svdshowmode that we can use to visualize the singular valuesand singular vectors of a nonsingular 2 2 matrix. Beforeusing the svdshow facility, we establish somebasic relations between the right and left singular vectors.