Solution to matrix equation PX=YP

Question

Consider three $n \times n$ real valued matrices, $P$, $X$, and $Y$ and the following equation:

$PX=YP$,

where $P$ is the unknown term.

Under what conditions is this equation solvable? Is it there a closed-form solution?

Your help is greatly appreciated.

Answer 1

Starting from the wikipedia entry for Sylvesters equation, which is:

$$PX-YP=C$$

it states:

Given complex $\displaystyle n\times n$ matrices $X$ and $Y$, Sylvester's equation has a unique solution $P$ for all $C$ if and only if $X$ and $Y$ have no common eigenvalues.

So, if $C=0$ and above criterion is fulfilled, $P=0$ is the only solution.

Furthermore, you can write the equation as

$$(I\otimes X - Y\otimes I)\text{vec}(P)=\text{vec}(C)$$

Hence, $P$ is in the null-space of $(I\otimes X - Y\otimes I)$.