How to solve this matrix equation

Question

Lets say I have this

$H = W*F*W'$

where all matrices are 2x2 and $F$ is a diagonal matrix.

$H$ and $F$ is known. Ho solve get $W$ in algebraic form?

I tried this here:

$inv(W)*H*inv(W') = F$ but it seems wrong... Its not homework.

Answer 1

Hint 1:

You can pre-multiply both sides of an equation by the same conformable term. So of you have

$$ X = AB $$ you can do $$ A^TX = A^TAB $$ or $$ XB^T = ABB^T $$

Hint 2:

If you have $X$ and you want to to go away, you can multiply it by $X^T(X^TX)^{-1}$

I suggest that you play around numerically to confirm the algebra. Here's something to get you started:

W = matrix(c(1,2,3,4),2)
f <- diag(3:4)
H <- W %*% f %*% t(W)