[Math] Matrices that are Hadamard products of $X$ and $X^{-T}$

Question

What are the matrices that you can write in the form $X \odot X^{-T}$, for a complex square matrix $X$, where $X^{-T}$ is the inverse of the complex transpose (not conjugate) and $\odot$ is the Hadamard (component-by-component) product?

In the $2\times 2$ case, you get the group of matrices in the form $$\begin{bmatrix}a & b\\\\ b & a \end{bmatrix},$$ with $a+b=1$, which are closed under matrix multiplication and would form a group were it not for the matrix $a=b=\frac12$ which admits no inverse [EDIT: corrected this assertion, thanks to Denis Serre]. In larger dimension, one sees that all the obtained matrices have the vector of all ones as both a right and left eigenvector. Is this the only restriction? Is the resulting set of matrices closed under multiplication? Is this problem known and studied?

Origin: motivated from this MO question.

Answer 1

There are some properties of this product in Horn and Johnson, "Topics in Matrix Analysis", Cambridge Univ Press 1991.