Solve for matrix that is hidden inside a scalar

Question

Let $X$ an invertible $n\times n$ matrix, parameter vectors $P, K$ $n\times 1$, vector $\Omega$, $1\times k$ and matrix $\Lambda$ $n\times k$. I would like to solve with respect to $X$ the equation: $$P^TX^{-1}KP^{T}X^{-1}\Lambda=\Omega$$ The gist of my question is how does one take out the unknown matrix from a scalar product.

Answer 1

(Edited)

I'll assume $\Omega \ne 0$. Since the left side is a linear combination of the rows of $\Lambda$, you need $\Omega$ to be such a linear combination, let's say $\Omega = R \Lambda$ where $R$ is $1 \times n$. You will also need $R K \ne 0$ ($> 0$ if you want a real solution). Now find an invertible matrix $Z$ so $P^T Z = R $. This is possible as long as $P \ne 0$. Now take $X = t Z^{-1}$ where $t$ is a suitable scalar. The equation becomes $t^{-2} (R K) \Omega = \Omega$, so you can take $t = (RK)^{-1/2}$.