# Eigendecomposition of the Hadamard product of a rank one symmetric matrix and a positive definite symmetric matrix

Is it possible to say anything about the eigenvalues and eigenvectors of a matrix

$X = Y \circ xx^T$

where $Y$ is a positive definite symmetric matrix with known eigen-decomposition

$Y=U\Lambda U^T$

and $x$ is a vector. In particular if

$X=V\Phi V^T$

what is the relationship between $V,U$ and $\Lambda, \Phi$. And/or is there an $O(n^2)$ to compute $V,\Phi$ from $U,\Lambda$ where $n$ is the size of the matrices.

• Note that $X = D Y D^T$, where $D=D^T$ is the diagonal matrix with the entries of $x$ on the diagonal -- this may be a more useful way to think about the problem. (That said, I don't know the answer.) – Federico Poloni Jul 18 '17 at 8:05