Doubly Stochastic Operators and the History of Birkhoff s Problem 111

Ex.4. Let ||tij|| be any matrix of non-negative terms from an ordered field, such that sums of the terms in each row and column is one. Show that it is a weighted mean of permutation matrices.

This statement is now referred to as Birkhoff's Theorem. Birkhoff gave a proof of it in Tres Observaciones sobre Álbegra Lineal [2].