Take a W is pxn matrix of rank n, $W^TW$ is invertible

$A = WSW^T$
A is pxp matrix with $rank\leq n$

$SW^T = (W^TW)^{-1} W^TA$

Taking Transpose

since S and A are symmetric

$WS = AW((W^TW)^{-1})^T$

$S = (W^TW)^{-1}W^TAW((W^TW)^{-1})^T$


$S^T = (W^TW)^{-1}W^TAW((W^TW)^{-1})^T$

If Rank of A is n

$D = Q^T (W^TW)^{-1}W^TAW((W^TW)^{-1})^T Q $

$D = Q^T (W^TW)^{-1}W^TP\Lambda P^TW((W^TW)^{-1})^T Q $

$S= P\Lambda P^T$ eigen value decomposition

A and S are symmetric matrix

Is there any mistake in my reasoning