Matrix equation solution

Question

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$

where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?

Answer 1

We could rephrase the whole equation as $P(1-\langle P,R\rangle) = X$ where $\langle a,b\rangle := a^t b $ is the standard scalar product. So $P$ is obviously a multiple of $X$ (note that the expression in the paranthesis is just a scalar), $P=tX$. Lets put that into the equation again:

$$tX (1- \langle tX,R\rangle) = X$$

$$\iff t(1-t\langle X,R\rangle) = 1$$

$$\iff -\langle X,R\rangle t^2+t-1 = 0 $$ which is a quadratic equation in one variable that can be solved for $t$. Then you have to plug in $t$ back into $P = tX$.

I hope this helps and everything is correct=)