$J=X^TPX$ , getting P from this formula

Question

I have a formula

$J=X^TPX$

P is n x n matrix

X is n x 1 matrix

J and X are known, how do I get P?

Answer 1

With a single $X$ and $J$, you can't. For instance $$ \begin{bmatrix} 1\\0\end{bmatrix} ^T\begin{bmatrix} a&b\\ c&d\end{bmatrix} \begin{bmatrix} 1\\0\end{bmatrix}=a. $$ You are completely free to choose $b,c,d$.

Conceptually, $J$ is a single number and $P$ is determined by $n^2$ numbers.