Generalized eigenvalue problem with real symmetric matrices and Hadamard product

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I have a generalized eigenvalue problem of the form

$$ (S \circ A) v = \lambda S v $$

where $\circ$ denotes the elementwise or Hadamard product and $S$ and $A$ are real symmetric matrices

As far as I know, there is no simple way to express the Hadamard product in terms of standard matrix multiplication, however it seems plausible that for this type of specifically constructed problem there is some simplification I don't know about as the matrix $S$ appears on both sides of the problem