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For some positive-definite symmetric matrix P, are there any known relationships between $A^T P A$ and $A P A^T$? (where $A$ may be non-invertible in general)

Update: Using the Matrix Inversion Lemma, I was able to show that $PA(P^{-1} + A^T P A)^{-1} = (P^{-1} + A P A^T)^{-1}AP$, but this is a bit more complicated than I was hoping. Still looking for other relationships and insights! Specifically, I am hoping for an explicit equation for $A^T P A$ in terms of $A P A^T$. Thanks in advance.

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    If $P$ is the identity matrix, do you expect an explicit equation for $A^TA$ in terms of $AA^T$? Try picking some $A$, doing both calculations, and see whether there's any relation. – 2017-02-13

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For example if $A$ is a symmetric matrix, then $A=A^T$, hence $A^TPA = APA^T$. I think it depends what type of matrix $A$ is.

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    If you write $A$ as the sum $A=A_S+A_A$ of its symetric ($A_S$) and antisymetric ($A_A$) components, then you get that $A^TPA=APA^T$ iff $A_SPA_A=A_APA_S$. This includes the cases $A_A=0$ (symetric) and $A_S=0$ (antisymetric). – 2017-02-13
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    Should have left that comment as an answer, thanks! – 2017-02-13