I have a system of n x 1$n \times 1$ equations
0 = A vec(xx') + B x + C
where
x is a n x 1 vector of unknowns
x' means transpose
vec means xx' has been vectorized so has dimension n^2 x 1
A is a known matrix with dimensions n x n^2
B is a known matrix with dimensions n x n $$ 0 = A\,vec(xx^t) + B x + C $$ where
C is a known vector of dimension n x 1
$x$ is a $n \times 1$ vector of unknowns
$x^t$ means transpose
$vec$ means $xx^t$ has been vectorized so has dimension $n^2 \times 1$
$A$ is a known matrix with dimensions $n \times n^2$
$B$ is a known matrix with dimensions $n \times n$
$C$ is a known vector of dimension $n \times 1$
I can solve these problems using a nonlinear solver. However, I am trying to find out if there are any theoretical results on how to solve this class of problems. I can find a lot of work on solving matrix quadratic equations. However, I can't find anything that has specifically this form, and I cannot figure out if I can rewrite this system in a way that is equivalent to other matrix quadratic problems I come across.
