For an arbitrary square matrix $A$, I'm looking for the set of vectors $x$ that $A$ maps orthogonal to $x$ (in other words, $x^T Ax = 0$). How can I go about solving this problem? What would I expect the solution set to look like (a hyperplane, maybe)?
Editing in some more info:
$x = 0$ is a solution, but it's not one I care about much. In general there aren't necessarily more solutions ($A = I$, for example), but I sort of lied to you: the form of $A$ is such that I expect at least one other.