The first two eigenvectors of A + t*E are almost indistinguishable indicating that the perturbed matrix is almost defective. If imag(d(1)) = 0, r = t else, s = t end endįinally, we display the perturbed matrix, which is obviously close to the original, and its pair of nearly equal eigenvalues. Now, a bisecting search, driven by the imaginary part of one of the eigenvalues, finds the point where two eigenvalues are nearly equal. With some trial and error which we do not show, we bracket the point where two eigenvalues of a perturbed A coalesce and then become complex. The direction of the required perturbation is given by E = -1.e-6*Y(:,1)*X(:,1)' We now proceed to show that A is close to a matrix with a double eigenvalue. In this example, the first eigenvalue has the largest sensitivity. It can be shown that perturbation of the elements of A can result in a perturbation of the j-th eigenvalue which is c(j) times as large. These three numbers are the reciprocals of the cosines of the angles between the left and right eigenvectors. We are now in a position to compute the sensitivities of the individual eigenvalues. More detailed information on the sensitivity of the individual eigenvalues involves the left eigenvectors. % (This is about the minimum condition that can be obtained by such diagonal scaling.) Moreover, it is now apparent that the three eigenvectors are nearly parallel. We can now forget how it was generated and analyze its eigenvalues. The similarity transformation is A = M*B/Mīecause det(M) = 1, the elements of A would be exact integers if there were no roundoff. The matrix M has determinant equal to 1 and is moderately badly conditioned. We now generate a similarity transformation to disguise the eigenvalues and make them more sensitive. (The value B(1,3) = 7 was chosen so that the elements of the matrix A below are less than 1000.) Moreover, since B is not symmetric, these eigenvalues are slightly sensitive to perturbation. Obviously, the eigenvalues of B are 1, 2 and 3. In this example, we construct a matrix whose eigenvalues are moderately sensitive to perturbations and then analyze that sensitivity.
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