I am having a problem to find the solution to the following equation which has arisen as part of the solution of a (convex) optimization problem I am considering. $$\left(\frac{a}{n \ln(\tau)}M^{-1}-x(\mu \bar{r}-\lambda e)^t\right)x=0$$ or rather, with appropriate constants, we can write this in general form $$(A-xc^t)x=0 $$ where $A$ (and $M$ of the original problem) is positive definite and $x,c \in \mathbb{R}^n$. I need to find $x$.
Also, I am interested in
$$(I-cx^t A)x=0$$
I would very much appreciate any help, references or much better a solution.
EDIT: Screwed my original post, just realized a tensor product xc^t can't be positive definite.
Thank you very much in advance.
For the first equation $(A - xc^t) x = 0$:
If $x$ is a solution of that equation, then we have $$ Ax = x (c^t x). $$ Apart from the trivial solution $x=0$, $x$ needs to be an eigenvector of $A$ and $c^t x$ the corresponding eigenvalue. Let $v$ be an eigenvector to eigenvalue $\lambda$. Then, we have $$ Av = \lambda v = (c^t v) v. $$ So $x = \frac{\lambda}{c^t v} v$ is a solution, if $c^t v \ne 0$.
For the second equation $(I - cx^t A) x = 0$:
If $x$ is a solution of that equation, then we have $$ x = c (x^t A x). $$ Apart from the trivial solution $x=0$, $x$ needs to be a multiple of $c$. Assume $x = \mu c$ for some $\mu\in\mathbb R\setminus\{0 \}$ and $c\ne 0$. Then, we have $$ \mu c = c \mu^2 (c^t A c). $$ Thus, $\mu = (c^t A c)^{-1}$ and $x = \frac{c}{c^t A c}$.