I need to find a real, symmetric matrix, $A$, that satisfies:
$\sum_{i,j} c_i c_j A_{ki}A_{jl} = A_{lk}$
I believe this is an equation of the form:
$c^T B c = A$, where $c$ is $\mathbb{R}^{N \times 1}$, $B$ is $\mathbb{R}^{N \times N \times N \times N}$ and $A$ is $\mathbb{R}^{N \times N}$
What, if any, is the method of solution to this problem, preferably analytic? I don't mind minimal additional assumptions to simplify it.
Thanks.
The equation can be written in matrix form (assuming $A=A^T$) as $ACC^TA=A$. Due to the rank bounds for products we have $$ \text{rank}A=\text{rank}ACC^TA\le\text{rank}C=1, $$ so the matrix has rank one or zero. $A=0$ is a trivial solution, so let's consider rank is one, i.e. $A=BB^T$ for some vector $B$. Set it into the equation $$ BB^T=BB^TCC^TBB^T\qquad\Leftrightarrow\qquad (B^TC)^2=1\qquad\Leftrightarrow\qquad B^TC=\pm 1. $$ Thus, all possible symmetric solutions are $A=0$ and $A=BB^T$ where $B^TC=\pm 1$.