I am searching for a way to reduce the following expression:
$$ \frac{1}{2}(y-\mu)^T\Sigma^{-1}(y-\mu) - x^TP^Ty= \frac{1}{2}(y-\tilde{\mu})^T\tilde{\Sigma}^{-1}(y-\tilde{\mu}) $$ I need to find a closed form expression of $\tilde{\mu}$ and $\tilde{\Sigma}$, I was tolde that this is possible by using normalization, but I have tried to complete the squares and some others things but it didn't help. I considered $\Sigma$ as an $n\times n$ diagonal matrix, and $P$ is a orthogonal matrix of the same dimensions, $x, y$ and $\mu$ are vectors of dimension $n \times 1$. I would appreciate any help or guidance with this. Thanks in advanzce.