Birkhoff-von Neumann Theorem

The Birkhoff-von Neumann Theorem states that any doubly stochastic matrix—one in which all entries are non-negative and each row and column sums to one—can be expressed as a mixture of permutation matrices, which are matrices representing perfect reordering or matching. Essentially, this means complex probabilistic arrangements can be broken down into simpler, deterministic reordering steps, providing a foundational link between randomness and permutation-based structures in mathematics and optimization.