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As the title says I have a large sparse matrix, 262000 by 262000, and i want to know how the number of times that an eigenvalue is degenerate. I can get the eigenvalue using the arnoldi method but after that i am clueless in what should i do. I can't calculate all of the eigenvalues of the matrix since i run out of memory.

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    $\begingroup$ Use Tally on the resulting list of eigenvalues? Since they are numerical, you may want to Round them to an appropriate precision first, to ease the comparison. $\endgroup$
    – MarcoB
    Commented May 23, 2023 at 17:00
  • $\begingroup$ The problem is: i can't calculate all of them without running out of memory. $\endgroup$
    – LittleBlue
    Commented May 23, 2023 at 17:03
  • $\begingroup$ That's an important part of your problem. Please edit your question to add that. $\endgroup$
    – MarcoB
    Commented May 23, 2023 at 17:08
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    $\begingroup$ Oh ok, I thought that was obvious since i said that was a large matrix. $\endgroup$
    – LittleBlue
    Commented May 23, 2023 at 17:22
  • $\begingroup$ provide minimal real example. just reduce your matrix and insert ur question. and what u tried with Large matrix that lead you to memory problem! $\endgroup$
    – nufaie
    Commented May 23, 2023 at 17:29

1 Answer 1

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If your memory is big enough to compute the orthogonal projector to the complement of each eigenvector and look for another eigenvalue of the same value in the projected matrix

   A . e = \lambda e
   P = TensorProduct[e, Transpose@Conjugate@e]
   (1-P) e = 0
   
   Solve[ (1-P).A.(1-P) . x   == lambda x,x]
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  • $\begingroup$ I very much doubt that symbolic methods will be of any use for a matrix this size. $\endgroup$ Commented May 24, 2023 at 11:44
  • $\begingroup$ This is a minutes test over some numbers. Depends on how sparse the matrix is. I tested it for a digaonal matrix and its unitary transformed with randoms of dimension 6. With these given, dimensions it looks like a social media on pairs problem of mid town size. $\endgroup$
    – Roland F
    Commented May 24, 2023 at 12:32

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