2 - Doubly stochastic matrices

Summary

A square matrix is called doubly stochastic if all entries of the matrix are nonnegative and the sum of the elements in each row and each column is unity. Among the class of nonnegative matrices, stochastic matrices and doubly stochastic matrices have many remarkable properties. Whereas the properties of stochastic matrices are mainly spectral theoretic and are motivated by Markov chains, doubly stochastic matrices, besides sharing such properties, also have an interesting combinatorial structure. In this chapter we first consider the combinatorial properties of the polytope of doubly stochastic matrices. The Birkhoff—von Neumann Theorem, the Frobenius-König Theorem, and related results are proved. An extension of the Frobenius-König Theorem involving matrix rank is given. We then describe a probabilistic algorithm to find a positive diagonal in a nonnegative matrix. Such algorithms are of relatively recent origin. The next several sections focus on diagonal products and permanents of nonnegative as well as doubly stochastic matrices. The proof of the van der Waerden conjecture due to Egorychev is given. We also give an elementary alternative proof of the Alexandroff Inequality, which is along the lines of the proof of the van der Waerden conjecture due to Falikman. The last few sections are concerned with various problems in game theory, scheduling, and economics.