Generalized EVP: right, left, and modified left eigenvectors for non-normal matrices

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BINDESH TRIPATHI

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Jun 11, 2021, 8:12:06 PM6/11/21

to dedalu...@googlegroups.com, keaton...@gmail.com, Adrian Fraser

Dear all,

First off, a really quick question: In the eigenvalue solver in Dedalus, is .A a shorthand for .toarray?

eig_output = eig(pencil.L_exp.A, b=-pencil.M_exp.A, **kw)

Next, I suppose, Dedalus constructs pencil.L_exp in Tz-spectral space by casting the original z-dependent linear operators (A and B here) in Chebyshev basis to solve A(z, d/dz)* X_i = λ_i * B(z, d/dz) * X_i (or AX = BXΛ). Does this seem like a correct understanding? If so, I believe, as an independent check to validate if all things are going well, I should be able to reconstruct the diagonal matrix (not necessarily Λ if the eigenvectors are not normalized properly) by computing, completely in Tz space, adjoint_of_Y * AX (where Y is a matrix of left eigenvectors whose each column is an eigenvector). Also, adjoint_of_Y * BX should be another diagonal matrix. Is that right?

Would the matrices A and B be the same as used in Dedalus if I construct them in the following manner? First, transform the output matrix (X) of right eigenvectors from Dedalus (each column is an eigenvector) from Tz-space to z-space, then apply the linear operator A(z,dz) on each column of X in z-space and then transform back each resulting column to Tz-space to form AX in Tz-space. Now, take the matrix multiplication of adjoint_of_Y (direct output from Dedalus) and AX (just constructed above), both in Tz-space, which I am imagining should be a diagonal matrix. However, when I do the same procedure in Dedalus, it does not yield a diagonal matrix. On the other hand, adjoint_of_modified_left_eigenvector*right eigenvector does produce a diagonal matrix in Tz-space. I am totally surprised why such a behavior manifests, given that, on a paper, adjoint_of_modified_left_eigenvector*right eigenvector is proportional to adjoint_of_Y * AX (or adjoint_of_Y * BX) and all of them should be diagonal. I believe Keaton has worked on similar problem before (and maybe many others here!). I would very much appreciate if I could learn from the Lords if I am missing anything trivial here. I have not given-up and am still overhauling all these matrices A, B, X, Y, modified Y, generated in Dedalus.