I have the following equation:
$$x^T M x = (x \circ b)^T P (x \circ b) $$
where
- $x, b \in \mathbb{R}^D$ are vectors
- $M, P \in \mathbb{R}^{D \times D}$ are matrices
- $b, P$ are known
- $\circ$ denotes the element-wise (Hadamard) product)
- The equation holds for all $x$.
How can I solve this equation for $M$?
If it helps, I know that M and P are both symmetric positive semi-definite matrices.
Best Answer
As WimC pointed out in the comments above, $x \cdot b$ can be written as $xB$ or $Bx$ where $B=diag(b)$. Consequently:
$$x^T M x = x^T B^T P B x$$
and therefore
$$M = B^T P B$$