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I have a real matrix $A \in \mathbb{R}^{n\times n}$ such that:

  • $A$ is symmetric
  • All the off-diagonal terms are known and positive
  • Has rank $k<n$

Unfortunately I don't know the values of the diagonal terms $A_{i,i}$, but I know that they are positive; what can be said in this case about SVD decomposition? Is there a way to calculate an approximate SVD having diagonal terms of the matrix to be decomposed undetermined?

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  • $\begingroup$ In which context does this arise? Any possibility of building some statistical or probability model for the missing diagonal? $\endgroup$
    – kjetil b halvorsen
    Commented Apr 10, 2016 at 9:29
  • $\begingroup$ If you could give more details of your real problem, I'm sure we can help! $\endgroup$
    – kjetil b halvorsen
    Commented Apr 10, 2016 at 9:56
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    $\begingroup$ If you can easily estimate off-diagonals, can you change basis and use this to estimate the diagonal? The opposite (polarization identities) is a well-known trick. $\endgroup$
    – Federico Poloni
    Commented Apr 10, 2016 at 11:54
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    $\begingroup$ This can be solved as an approximate optimization problem, without truly doing a true SVD. $\endgroup$
    – Suvrit
    Commented Apr 10, 2016 at 14:36
  • $\begingroup$ @Suvrit and Federico, do you have a reference for your comments? I would be great! $\endgroup$
    – Ulderique Demoitre
    Commented Apr 10, 2016 at 15:33

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