I'm trying to find information on the eigenvalues of an n×n matrix $A$ such that

$A=D+J$ Where $D$ is some complex valued diagonal matrix, and $J$ is a rank one matrix, $J = uu^T$. How to compute the eigen values of $A$ in this case?

  • 4
    $\begingroup$ Perhaps this might be useful: Bunch–Nielsen–Sorensen formula? $\endgroup$ Apr 29, 2018 at 15:33
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    $\begingroup$ Questions about eigenvalues after rank one updates have been asked several times on MO previously; please search for them; most likely they already answer your question. $\endgroup$
    – Suvrit
    Apr 29, 2018 at 16:40
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    $\begingroup$ Just to avoid possible missunderstandings: do you assume the perturbation $J$ to be a general rank-$1$ matrix (which would be of the form $J = uv^T$ rather than $J = uu^T$), or do you really wish $J$ to be of the form $J = uu^T$ (and thus symmetric)? $\endgroup$ Apr 29, 2018 at 19:35
  • $\begingroup$ Yes, I assume $J = uu^T$ $\endgroup$
    – Christo
    Jul 26, 2018 at 10:19
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    $\begingroup$ My question above is not answered by any of the previous questions asked here. Since the previous questions here assume D to be an identity matrix and my case D being diagonal with all entries distinct. Could someone have an answer for this case? $\endgroup$
    – Christo
    Aug 8, 2018 at 13:24


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