I have the following matrix:
$$A_{ij} = B_{ij} C_{ij} = v_i v_j C_{ij}$$
where $v$ is a vector of wavenumbers and $C$ is a circulant matrix. I want to find the eigenvalues/vectors of $A$. The matrix $B$ is of rank 1 with eigenvalue $\lambda^B = \sum_i v_i^2$, and the eigenvalues/vectors of $C$ are given by the (inverse) Fourier transform (see Wikipedia).
Is there any way of relating the eigenvalues/vectors of $A$ to those of $B$ and $C$, in this particular case?
Now $$ A = DCD $$ where $D = diag(v_1\dots v_n)$. I have no idea how to get the eigenvalues of $A$ from those of $C$.
[Answer for first version of question] The matrix $A$ satisfies $$ A = v v^TC. $$ It has rank at most $1$, thus minimum $n-1$ zero eigenvalues. If $v^TC=0$, then $A=0$. If $v^TCv=0$ then $A^2=0$, and $A$ is nilpotent and has no non-zero eigenvalues. If $v^TCv\ne0$, then this value is an eigenvalue to the eigenvector $v$: $$ Av = vv^TCv = (v^TCv) v. $$