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I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g., to do optimisation over them.

I have seen quite a few examples of similar structures, like the cone/manifold of positive semidefinite (PSD) matrices, the fixed rank manifold, the symmetric positive semidefinite fixed-ranked manifold, etc.

I am not so familiar with the literature; perhaps someone can share their knowledge on the possible geometry of symmetric indefinite matrices of fixed rank?

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  • $\begingroup$ You might want to take a look at slides 13-19 of Parrilo's The convex algebraic geometry of rank minimization (2009) [PDF] $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 20, 2023 at 11:33
  • $\begingroup$ Thank you for your reply! I am aware of the fixed-rank geometry, but I was referring to the restriction to symmetric matrices of this manifold. Specifically, is there a nice characterisation of the restriction of fixed-rank matrices to only the symmetric ones? $\endgroup$
    – turtlesandwich
    Commented Dec 25, 2023 at 22:59
  • $\begingroup$ Have you taken a look at Optimization algorithms on matrix manifolds? $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 25, 2023 at 23:28
  • $\begingroup$ Have you considered starting with symmetric rank-1 matrices? $\endgroup$
    – Rodrigo de Azevedo
    Commented Dec 25, 2023 at 23:31

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