I'm looking for help in finding solutions for $| x \rangle$ in the following equation: $$ \langle x | M | x \rangle = \langle y | x \rangle. $$ The vectors $| x \rangle$ and $|y \rangle$ are in general complex-valued vectors. In my notation (Dirac notation) $| x \rangle$ stands for the complex transpose of $\langle x |$.
As posed, we are given $M$ and $| y \rangle$. $M$ (a real matrix in this case) can be broken into its symmetric and antisymmetric parts, $M = A + S$. Since $\langle x | A | x \rangle = 0$, we have $$ \langle x | S | x \rangle = \langle y | x \rangle. $$ I'm not sure if this helps, but it simplifies the problem a little.
The question is: is there any way to say anything concrete about the solutions $| x \rangle$? I expect that the solutions will be degenerate, and that is okay. However I have never seen an equation like this and it's throwing me for a loop.
Thanks for any guidance.