Vtyjkyuk

(This question is part of my larger quest to solve the question here: Solving simultaneous quadratic equations ("almost" linear...), arising from maximum likelihood estimation problem)

Suppose I have a quadratic vector equation of the form

$mathrmdiag(a) mathrmdiag(x) x + B x + c = 0 quad (1)$

Where $mathrmdiag(v)$ indicates the diagonal matrix having the vector $v$ on the diagonal, B is an invertible square matrix, and x and c are vectors. Is this solvable for $x$?

As part of my attempt, I noticed that this could be written as a quadratic form by "dotting" it with an arbitrary vector $v$:

$v^T mathrmdiag(a) mathrmdiag(x) x + v^T B x + v^T c = 0$

which (I'm pretty sure) can be rewritten as:

$x^T mathrmdiag(a) mathrmdiag(v) x + v^T B x + v^T c = 0$

which, if one completes the square, is the equation for an ellipsoid/hyperboloid. So I think this means that $x$ lies on an ellipsoid type surface. But, $(1)$ is much more restrictive than this, so I am hoping that it is possible to find precisely which point on the ellipsoid it describes.

Am I on the right track? Or is this clearly not possible?

Edit: I think this can be written as a quadratic matrix equation for $mathrmdiag(x)$:

$mathrmdiag(x) mathrmdiag(x) a + B mathrmdiag(x) 1 + c = 0$

which according to Is there a unique solution for this quadratic matrix equation? isn't solvable in general, but maybe it is solvable when the matrix is diagonal?