Birkhoff–von Neumann theorem states that a polytope formed by a set of doubly-stochastic matrices has extreme points that are permutation matrices.


I am wondering if there is a similar theorem for a permutation matrix whose elements are replaced by some values of $e^{ix}$.



"Permutation matrix" but non-zero entries are replaced by $e^{ix}$
A statement similar to the Birkhoff theorem holds: these matrices are the extreme points of the set of matrices for which the $\ell\_1$ norm of each row and column is $\leq 1$. I denote by $K$ this set.


It is clear that every such matrix is an extreme point in $K$. Conversely, let $A$ be an extreme point in $K$. The $\ell\_1$ norm of each row and column must be $1$. Write the entrywise polar decomposition, i.e. write $A$ as an entrywise product $A=U \circ B$ where $B$ is a bistochastic matrix and $U$ a matrix with entries having modulus $1$. The matrix $B$ is an extremal bistochastic matrix: if $B=(B\_1 + B\_2)/2$ for bistochastic matrices $B\_1 \neq B\_2$, then the equation $A = (U \circ B\_1+U\circ B\_2)/2$ contradicts the extremality of $A$. By the usual Birkhoff theorem, $B$ is a permutation matrix and the result follows.


