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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 $$\begin{bmatrix}a & b\\ b & a \end{bmatrix},end{bmatrix},$$ with $a+b=1$, which are closed under matrix multiplication and would form a group were it not for the hyperbolic rotations matrix $\oplus$ a multiplicative factora=b=\frac12$which admits no inverse [EDIT: corrected this assertion, thanks to Denis Serre]. In larger dimension, one sees that all the obtained matrices have the vector of all ones as both a right and left eigenvector. Is this the only restriction? Do Is the resulting set of matrices that we obtain form a groupclosed under multiplication? Is this problem known and studied?  Origin: motivated from this MO question.        1 # Matrices that are Hadamard products of$X$and$X^{-T}$What are the matrices that you can write in the form$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}, which are the hyperbolic rotations$\oplus\$ a multiplicative factor. 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? Do the matrices that we obtain form a group? Is this problem known and studied?

Origin: motivated from this MO question.