On complex matrix scalings of extremal permanent

A doubly quasi-stochastic (DQS) matrix is said to be maximally (minimally) scaled if it cannot be diagonally scaled to another doubly quasi-stochastic matrix with larger (smaller) permanent. Motivated by a connection to the geometric measure of entanglement of certain symmetric states, we offer a series of results on the structures of the sets of n×n maximally scaled (MaxScn) and minimally scaled (MinScn) DQS matrices. In particular, we offer a characterization of the set of n×n maximally scaled matrices, and use this characterization to show that these matrices form a convex set and that the n×n identity matrix is the element of MaxScn with smallest permanent. We then show that real DQS matrices in MaxScn or MinScn must satisfy certain spectral properties, and use these properties to show that all positive definite doubly stochastic matrices are minimally scaled. We finish with a bound on the permanent of any real matrix or Abelian group matrix in MinScn.