With each A Rnn, we can associate n closed circular disks in the complex plane. The ith

Chapter 7, Problem 16

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With each A Rnn, we can associate n closed circular disks in the complex plane. The ith disk is centered at aii and has radius ri = _n j=1 j _=i |ai j | Each eigenvalue of A is contained in at least one of the disks (see Exercise 7 of Section 6). (a) Set A = round(10 rand(5)) Compute the radii of the Gerschgorin disks of A and store them in a vector r. To plot the disks, we must parameterize the circles. This can be done by setting t = [0 : 0.1 : 6.3]_; We can then generate two matrices X and Y whose columns contain the x and y coordinates of the circles. First, we initialize X and Y to zero by setting X = zeros(length(t), 5); Y = X; The matrices can then be generated with the following commands: for i = 1 : 5 X(:, i ) = r (i ) cos(t) + real(A(i, i )); Y (:, i ) = r (i ) sin(t) + imag(A(i, i )); End and the disks with the command plot(X, Y, real(e), imag(e), x) If everything is done correctly, all the eigenvalues of A should lie within the union of the circular disks. (b) If k of the Gerschgorin disks form a connected domain in the complex plane that is isolated from the other disks, then exactly k of the eigenvalues of the matrix will lie in that domain. Set B = [3 0.1 2; 0.1 7 2; 2 2 50] (i) Use the method described in part (a) to compute and plot the Gerschgorin disks of B. (ii) Since B is symmetric, its eigenvalues are all real and so must all lie on the real axis. Without computing the eigenvalues, explain why B must have exactly one eigenvalue in the interval [46, 54]. Multiply the first two rows of B by 0.1 and then multiply the first two columns by 10. We can do this in MATLAB by setting D = diag([0.1, 0.1, 1]) and C = D B/D The new matrix C should have the same eigenvalues as B. Why? Explain. Use C to find intervals containing the other two eigenvalues. Compute and plot the Gerschgorin disks for C. (iii) How are the eigenvalues of CT related to the eigenvalues of B and C? Compute and plot the Gerschgorin disks for CT . Use one of the rows of CT to find an interval containing the largest eigenvalue of CT . 1