For various values of k, form a k k matrix A by setting D = diag(ones(k 1, 1), 1); A = 2

The top matrix on the menu is the diagonal matrix A = 5 4 0 0 3 4 Initially, when you select this matrix, the vectors x and Ax should both be aligned along the positive x-axis. What information about an eigenvalue eigenvector pair is apparent from the initial figure positions? Explain. Rotate x counterclockwise until x and Ax are parallel, that is, until they both lie along the same line through the origin. What can you conclude about the second eigenvalue eigenvector pair? Repeat this experiment with the second matrix. How can you determine the eigenvalues and eigenvectors of a 2 2 diagonal matrix by inspection without doing any computations? Does this also work for 3 3 diagonal matrices? Explain.

The third matrix on the menu is just the identity matrix I. Howdo x and I x compare geometrically as you rotate x around the unit circle? What can you conclude about the eigenvalues and eigenvectors in this case?

The fourth matrix has 0s on the diagonal and 1s in the off-diagonal positions. Rotate the vector x around the unit circle and note when x and Ax are parallel. On the basis of these observations, determine the eigenvalues and the corresponding unit eigenvectors. Check your answers by multiplying the matrix times the eigenvector to verify that Ax = x.

The next matrix in the eigshow menu looks the same as the previous ones, except that the (2, 1) entry has been changed to 1. Rotate the vector x completely around the unit circle. Are x and Ax ever parallel? Does A have any real eigenvectors? What can you conclude about the nature of the eigenvalues and eigenvectors of this matrix?

Investigate the next three matrices on the menu (the sixth, seventh, and eighth). In each case, try to estimate geometrically the eigenvalues and eigenvectors and make your guesses for the eigenvalues consistent with the trace of the matrix. Use MATLAB to compute the eigenvalues and eigenvectors of the sixth matrix by setting [X, D] = eig([0.25, 0.75 ; 1, 0.50 ]) The column vectors of X are the eigenvectors of the matrix and the diagonal entries of D are the eigenvalues. Check the eigenvalues and eigenvectors of the other two matrices in the same way.

Investigate the ninth matrix on the menu. What can you conclude about the nature of its eigenvalues and eigenvectors? Check your conclusions by computing the eigenvalues and eigenvectors with the eig command.

Investigate the next three matrices on the menu. You should note that, for the last two of these matrices, the two eigenvalues are equal. For each matrix, how are the eigenvectors related? Use MATLAB to compute the eigenvalues and eigenvectors of these matrices.

The last item on the eigshow menu will generate a random 2 2 matrix each time that it is invoked. Try using the random matrix 10 times, and in each case determine whether the eigenvalues are real. What percentage of the 10 random matrices had real eigenvalues? What is the likelihood that two real eigenvalues of a random matrix will turn out to be exactly equal? Explain.

Consider the application relating to critical loads for a beam from Section 1. For simplicity, we will assume that the beam has length 1 and that its flexural rigidity is also 1. Following the method described in the application, if the interval [0, 1] is partitioned into n subintervals, then the problem can be translated into a matrix equation Ay = y. The critical load for the beam can be approximated by setting P = sn2, where s is the smallest eigenvalue of A. For n = 100, 200, 400, form the coefficient matrix by setting D = diag(ones(n 1, 1), 1); A = eye(n) D D_ In each case, determine the smallest eigenvalue of A by setting s = min(eig(A)) and then compute the corresponding approximation to the critical load. 1

Construct a symmetric matrix A by setting A = round(5 rand(6)); A = A + A_ Compute the eigenvalues of A by setting e = eig(A). (a) The trace of A can be computed with the MATLAB command trace(A), and the sum of the eigenvalues of A can be computed with the command sum(e). Compute both of these quantities and compare the results. Use the command prod(e) to compute the product of the eigenvalues of A and compare the result with det(A). (b) Compute the eigenvectors of A by setting [X, D] = eig(A). Use MATLAB to compute X1AX and compare the result with D. Compute also A1 and XD1X1 and compare the results. 1

Set A = ones(10) + eye(10) (a) What is the rank of A I ? Why must = 1 be an eigenvalue of multiplicity 9? Compute the trace of A, using the MATLAB function trace. The remaining eigenvalue 10 must equal 11. Why? Explain. Compute the eigenvalues of A by setting e = eig(A). Examine the eigenvalues, using format long. How many digits of accuracy are there in the computed eigenvalues? (b) The MATLAB routine for computing eigenvalues is based on the QR algorithm described in Section 6 of Chapter 7. We can also compute the eigenvalues of A by computing the roots of its characteristic polynomial. To determine the coefficients of the characteristic polynomial of A, set p = poly(A). The characteristic polynomial of A should have integer coefficients. Why? Explain. If we set p = round(p), we should end up with the exact coefficients of the characteristic polynomial of A. Compute the roots of p by setting r = roots(p) and display the results, using format long. How many digits of accuracy are there in the computed results? Which method of computing eigenvalues is more accurate, using the eig function or computing the roots of the characteristic polynomial? 1

Consider the matrices A = 5 3 3 5 and B = 5 3 3 5 Note that the two matrices are the same except for their (2, 2) entries. (a) Use MATLAB to compute the eigenvalues of A and B. Do they have the same type of eigenvalues? The eigenvalues of the matrices are the roots of their characteristic polynomials. Use the followingMATLAB commands to form the polynomials and plot their graphs on the same axis system: p = poly(A); q = poly(B); x = 8 : 0.1 : 8; y = polyval(p, x); z = zeros(size(x)); w = polyval(q, x); plot(x, y, x,w, x, z) hold on The hold on command is used so that subsequent plots in part (b) will be added to the current figure. How can you use the graph to estimate the eigenvalues of A? What does the graph tell you about the eigenvalues of B? Explain. (b) To see how the eigenvalues change as the (2, 2) entry changes, let us construct a matrix C with a variable (2, 2) entry. Set t = sym( _t_ ) C = [5,3; 3, t 5] As t goes from 0 to 10, the (2, 2) entries of these matrices go from 5 to 5. Use the following MATLAB commands to plot the graphs of the characteristic polynomials for the intermediate matrices corresponding to t = 1, 2, . . . , 9: p = poly(C) for j = 1 : 9 s = subs(p, t, j ); ezplot(s, [10, 10]) axis([10, 10,20, 220]) pause(2) end Which of these intermediate matrices have real eigenvalues and which have complex eigenvalues? The characteristic polynomial of the symbolic matrix C is a quadratic polynomial whose coefficients are functions of t. To find exactly where the eigenvalues change from real to complex, write the discriminant of the quadratic as a function of t and then find its roots. One root should be in the interval (0, 10). Plug that value of t back into the matrix C and determine the eigenvalues of the matrix. Explain how these results correspond to your graph. Solve for the eigenvectors by hand. Is the matrix diagonalizable? 1

Set B = toeplitz(0: 1: 3, 0: 3) The matrix B is not symmetric and hence it is not guaranteed to be diagonalizable. Use MATLAB to verify that the rank of B equals 2. Explain why 0 must be an eigenvalue of B and the corresponding eigenspace must have dimension 2. Set [X, D] = eig(B). Compute X1BX and compare the result with D. Compute also XD5X1 and compare the result with B5. 1

Set C = triu(ones(4), 1) + diag([1, 1],2) and [X, D] = eig(C) Compute X1CX and compare the result with D. Is C diagonalizable? Compute the rank of X and the condition number of X. If the condition number of X is large, the computed values for the eigenvalues may not be accurate. Compute the reduced row echelon form of C. Explain why 0 must be an eigenvalue of C and the corresponding eigenspace must have dimension 1. Use MATLAB to compute C4. It should equal the zero matrix. Given that C4 = O, what can you conclude about the actual values of the other three eigenvalues of C? Explain. Is C defective? Explain. 1

Construct a defective matrix by setting A = ones(6); A = Atril(A)triu(A, 2) It is easily seen that = 0 is the only eigenvalue of A and that its eigenspace is spanned by e1. Verify that this is indeed the case by using MATLAB to compute the eigenvalues and eigenvectors of A. Examine the eigenvectors, using format long. Are the computed eigenvectors multiples of e1? Now perform a similarity transformation on A. Set Q = orth(rand(6)); and B = Q_ A Q If the computations had been done in exact arithmetic, the matrix B would be similar to A and hence defective. Use MATLAB to compute the eigenvalues of B and a matrix X consisting of the eigenvectors of B. Determine the rank of X. Is the computed matrix B defective? Because of rounding error, a more reasonable question to ask is whether the computed matrix B is close to being defective (i.e., are the column vectors of X close to being linearly dependent?). To answer this question, use MATLAB to compute rcond(X), the reciprocal of the condition number of X. A value of rcond close to zero indicates that X is nearly rank deficient. 1

Generate a matrix A by setting B = [1, 1; 1, 1 ], A = [zeros(2), eye(2); eye(2), B] (a) The matrix A should have eigenvalues 1 = 1 and 2 = 1. Use MATLAB to verify that these are the correct eigenvalues by computing the reduced row echelon forms of A I and A + I . What are the dimensions of the eigenspaces of 1 and 2? (b) It is easily seen that trace(A) = 0 and det(A) = 1. Verify these results in MATLAB. Use the values of the trace and determinant to prove that 1 and 1 are actually both double eigenvalues. Is A defective? Explain. (c) Set e = eig(A) and examine the eigenvalues, using format long. How many digits of accuracy are there in the computed eigenvalues? Set [X, D] = eig(A) and compute the condition number of X. The log of the condition number gives an estimate of how many digits of accuracy are lost in the computation of the eigenvalues of A. (d) Compute the rank of X. Are the computed eigenvectors linearly independent? Use MATLAB to compute X1AX. Does the computed matrix X diagonalize A? 1

Suppose that 10,000 men and 10,000 women settle on an island in the Pacific that has been opened to development. Suppose also that a medical study of the settlers finds that 200 of the men are color blind and only 9 of the women are color blind. Let x(1) denote the proportion of genes for color blindness in the male population and let x(2) be the proportion for the female population. Assume that x(1) is equal to the proportion of color-blind males and that x(2)2 is equal to the proportion of color-blind females. Determine x(1) and x(2) and enter them in MATLAB as a column vector x. Enter also the matrix A from Application 3 of Section 3. Set MATLAB to format long, and use the matrix A to compute the proportions of genes for color blindness for each sex in generations 5, 10, 20, and 40. What are the limiting percentages of genes for color blindness for this population? In the long run, what percentage of males and what percentage of females will be color blind? 1

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 1

Construct a complex Hermitian matrix by setting j = sqrt(1); A = rand(5) + j rand(5); A = (A + A_ )/2 (a) The eigenvalues of A should be real. Why? Compute the eigenvalues and examine your results, using format long. Are the computed eigenvalues real? Compute also the eigenvectors by setting [X, D] = eig(A) What type of matrix would you expect X to be? Use the MATLAB command X_ X to compute XHX. Do the results agree with your expectations? (b) Set E = D + j eye(5) and B = X E/X What type of matrix would you expect B to be? Use MATLAB to compute BHB and BBH . How do these two matrices compare? 2

Let A be a nonsingular 2 2 matrix with singular value decomposition A = USVT and singular values s1 = s11 and s2 = s22. Explain why each of the following are true: (a) AV = US (b) Av1 = s1u1 and Av2 = s2u2. (c) v1 and v2 are orthogonal unit vectors and the images Av1 and Av2 are also orthogonal. (d) _Av1_ = s1 and _Av2_ = s2. 2

Set A = [1, 1; 0.5,0.5] and use MATLAB to verify each of statements (a)(d) in Exercise 20. Use the command eigshow(A) to apply the eigshow utility to the matrix A. Click on the eig/(svd) button to switch into the svdshow mode. The display in the figure window should show a pair of orthogonal vectors x, y and their images Ax and Ay. Initially, the images of x and y should not be orthogonal. Use the mouse to rotate the x and y vectors counterclockwise until their images Ax and Ay become orthogonal. When the images are orthogonal, x and y are right singular vectors of A. When x and y are right singular vectors, how are the singular values and left singular vectors related to the images Ax and Ay? Explain. Note that when you rotate a full 360, the image of the unit circle traces out as an ellipse. How do the singular values and singular vectors relate to the axes of the ellipse? 2

Use the following MATLAB commands to construct a symbolic function: syms x y f = (y + 1)3 + x y2 + y2 4 x y 4 y + 1 Compute the first partials of f and the Hessian of f by setting f x = diff( f, x), f y = diff( f, y) H = [diff( f x, x), diff( f x, y); diff( f y, x), diff( f y, y)] We can use the subs command to evaluate the Hessian for any pair (x, y). For example, to evaluate the Hessian when x = 3 and y = 5, set H1 = subs(H,[x,y],[3,5]) Use the MATLAB command solve( f x, f y) to determine vectors x and y containing the x and y coordinates of the stationary points. Evaluate the Hessian at each stationary point and then determine whether the stationary point is a local maximum, local minimum, or saddle point. 2

Set C = ones(6) + 7 eye(6) and [X, D] = eig(C) (a) Even though = 7 is an eigenvalue of multiplicity 5, the matrix C cannot be defective. Why? Explain. Check that C is not defective by computing the rank of X. Compute also XTX. What type of matrix is X? Explain. Compute also the rank of C7I . What can you conclude about the dimension of the eigenspace corresponding to = 7? Explain. (b) The matrix C should be symmetric positive definite. Why? Explain. Thus, C should have a Cholesky factorization LLT . The MATLAB command R = chol(C) will generate an upper triangular matrix R that is equal to LT . Compute R in this manner and set L = R_. Use MATLAB to verify that C = LLT = RT R (c) Alternatively, one can determine the Cholesky factors from the LU factorization of C. Set [L U] = lu(C) and D = diag(sqrt(diag(U))) and W = (L D) _ How do R and W compare? This method of computing the Cholesky factorization is less efficient than the method MATLAB uses for its Chol function. 2

For various values of k, form a k k matrix A by setting D = diag(ones(k 1, 1), 1); A = 2 eye(k) D D_; In each case, compute the LU factorization of A and the determinant of A. If A is an n n matrix of this form, what will its LU factorization be? What will its determinant be? Why must the matrix be positive definite? 2

For any positive integer n, the MATLAB command P = pascal(n) will generate an n n matrix P whose entries are given by pi j = _ 1 if i = 1 or j = 1 pi1, j + pi, j1 if i > 1 and j > 1 The name pascal refers to Pascals triangle, a triangular array of numbers that is used to generate binomial coefficients. The entries of the matrix P form a section of Pascals triangle. (a) Set P = pascal(6) and compute the value of its determinant. Now subtract 1 from the (6, 6) entry of P by setting P(6, 6) = P(6, 6) 1 and compute the determinant of the new matrix P. What is the overall effect of subtracting 1 from the (6, 6) entry of the 6 6 Pascal matrix? (b) In part (a) we saw that the determinant of the 6 6 Pascal matrix is 1, but if we subtract 1 from the (6, 6) entry, the matrix becomes singular. Will this happen in general for n n Pascal matrices? To answer this question, consider the cases n = 4, 8, 12. In each case, set P = pascal(n) and compute its determinant. Next, subtract 1 from the (n, n) entry and compute the determinant of the resulting matrix. Does the property that we discovered in part (a) appear to hold for Pascal matrices in general? (c) Set P = pascal(8) and examine its leading principal submatrices. Assuming that all Pascal matrices have determinants equal to 1, why must P be positive definite? Compute the upper triangular Cholesky factor R of P. How can the nonzero entries of R be generated as a Pascal triangle? In general, how is the determinant of a positive definite matrix related to the determinant of one of its Cholesky factors? Why must det(P) = 1? (d) Set R(8, 8) = 0 and Q = R_ R The matrix Q should be singular. Why? Explain. Why must the matrices P and Q be the same except for the (8, 8) entry? Why must q88 = p88 1? Explain. Verify the relation between P and Q by computing the difference P Q.