Universality for Doubly Stochastic Matrices

TL;DR

This paper proves that no finite set of doubly stochastic matrices can generate all possible entries, highlighting a fundamental difference from stochastic or unitary matrices, with implications for matrix universality and topological semigroup theory.

Contribution

It establishes the non-existence of a finite universal set for doubly stochastic matrices using topological semigroup concepts, contrasting with known results for other matrix classes.

Findings

Finite sets of doubly stochastic matrices generate a nowhere dense set.

No finite universal set exists for doubly stochastic matrices.

The proof employs a theorem on topological semigroups with the convergent property.

Abstract

We show that the set of entries generated by any finite set of doubly stochastic matrices is nowhere dense, in contrast to the cases of stochastic matrices or unitary matrices. In other words, there is no finite universal set of doubly stochastic matrices, even with the weakest notion of universality. Our proof is based on a theorem for topological semigroups with the convergent property. A topological semigroup is convergent if every infinite product converges. We show that in a compact and convergent semigroup, under some restrictions, the closure of every finitely generated subsemigroup can be instead generated directly by the generating set and the limits of infinite products.