This saddle point was found in search # 1ĭistance to nearest defective matrix B found is 5.01214e-07 Neardefmat: lower bound on distance is 3.4791e-07 and upper bound is 8.76938e-05 The red contour shows the regions containing the two that are coalescing.įind the nearest defective matrix. We expect the nearest defective matrix to bring two of them together.įocus on the three smallest eigenvalues. The three smallest eigenvalues have the largest condition numbers. format longĪnd here are the corresponding condition numbers. Since it is almost upper triangular you might think the eigenvalues are near the diagonal elements, but they are not. The matrix is clearly far from symmetric. The 9x9 Frank matrix fits nicely on the page. Ill-conditioned eigenvalues-the smaller ones. They are positive and occur in reciprocal pairs Of F may be obtained in terms of the zeros of the Hermite Reflected about the anti-diagonal (1,N)-(N,1). help private/frank FRANK Frank matrix.į = GALLERY('FRANK',N, K) is the Frank matrix of order N. Of course, we can find them in the gallery. Wilkinson was interested in the Frank family of matrices. The size of the perturbation is sigma = S(3,3)īy construction, B has a double eigenvalue, but the eigenvalue's condition is infinite, so it is not computed accurately. This is essentially the same perturbation that I used to construct A in last week's blog post. Neardefmat: two closest computed eigenvalues of B differ by 0.00269716Īnd condition numbers of these eigenvalues are 272230, 272669 This saddle point was found in search # 3ĭistance to nearest defective matrix B found is 0.000375171 Neardefmat: lower bound on distance is 0.000299834 and upper bound is 0.00618718 By default neardefmat prints some useful details about the computation. The outputs from neardefmat are B, the nearest defective matrix, and z, the resulting double eigenvalue. To see more precisely where they coalesce, we use the function neardefmat, nearest defective matrix, written by Michael Overton, a professor at NYU. This animated gif adjusts the region around the eigenvalues at 2 and 3 until the regions coalesce in a saddle point near 2.4. The contours outline the regions where the eigenvalues can move when the matrix is perturbed by a specified amount. You can see the three eigenvalues at 1, 2 and 3. Here is the pseudospectrum of our example matrix. It was developed at Oxford from 1999 - 2002 by Thomas Wright under the direction of Nick Trefethen. kappa = (vecnorm(Y').*vecnorm(X))'ĮigTool is open MATLAB software for analyzing eigenvalues, pseudospectra, and related spectral properties of matrices. Y*XĪ function introduced in 2017b that computes the 2-norm of the columns of a matrix comes in handy at this point. They are normalized so that their dot product with the right eigenvectors is one. These condition numbers come from the right eigenvector matrix = eig(A)Īnd the left eigenvector matrix Y = inv(X) This is predicted by the eigenvalue condition numbers, format short For a moment, let's forget how it was constructed and start with A = In last week's blog post, "An Eigenvalue Sensitivity Example", I resurrected this example from the 1980 User's Guide.
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