Solve an matrix trace equation: find x in Tr[(Y-xWY-FB)'(WY)]=0

Question

I have the following linear equation

$$ \mathrm{Tr}[(Y-xWY-FB)'(WY)]=0$$

where $Y$ is an $n$-by-$b$ matrix, $W$ is an $n$-by-$n$ matrix, $F$ is an $n$-by-$m$ matrix, $B$ is an $m$-by-$b$ matrix, and $x$ is an scalar. All $Y$, $W$, $F$ and $B$ matrices are known, but $x$ is unknown. Is there a close-form solution for $x$? Thank you all in advance.

Answer 1

Since the trace is linear, so you can easily isolate x:

$$ \eqalign { x*Tr[(WY)'(WY)] &= Tr[(Y-FB)'(WY)] \cr x &= \frac{Tr[(Y-FB)'(WY)] }{ Tr[(WY)'(WY)]} \cr } $$