SOME RESULTS ON NON-NEGATIVE MATRICES

Recently Mirsky and Farahat proposed the problem of characterizing the class of doubly stochastic mat riees for which t he least number of permutation matrices necessary to r epresent it as a convex sum has a prescribed va lu e. It is shown that this number can be related t o t he number of eigenvalues of modulus one. The problem of similarity of doubly stochastic matrices is also treated . Finally, the question of t ransitivity of powers of sets of functions on t he first n positive integers into itself is treated by defining a corresponding incidence mat rix and exa minin g its powers .