The Birkhoff theorem for unitary matrices of arbitrary dimensions

TL;DR

This paper generalizes Birkhoff's theorem for unitary matrices with equal line sums from prime dimensions to arbitrary dimensions, broadening its applicability in matrix theory.

Contribution

It provides a comprehensive proof extending the Birkhoff theorem for unitary matrices to all dimensions, not limited to prime numbers.

Findings

The theorem now applies to any matrix dimension.

The proof covers both prime and composite dimensions.

This advances the theoretical understanding of unitary matrices with equal line sums.

Abstract

It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.