Getting the original matrix from the projection

Question

If I have the projection matrix of $X$, $$ P = X\,{(X^TX)}^{-1} X^T, $$ how can I recover $X$ by only knowing $P$? Is there a way to do that?

Answer 1

Multiplying $\,P$ by $X\,$ yields $$\eqalign{ PX &= X(X^TX)^{-1}X^TX \;\doteq\; X \\ }$$ Therefore, given any random matrix $A:\quad X=PA$
$\sf Proof:$ $$\eqalign{ PX &= P\left(PA\right) \,=\, PA \;\doteq\; X \quad \\ }$$ So you can certainly use $P$ to calculate $X,\,$ however the solution isn't unique
because every $A$ matrix leads to a different $X$.