Real essentially stochastic matrices: factorizations into special elementary matrices

In this paper we show that every real essentially stochastic (r.e.s.) matrix A of order n ≥ 2 is a product of at most q(n) elementary r.e.s. matrices with positive determinants (i.e., positive main diagonal entries), plus a transposition matrix if det A < 0, where q( n) is a quadratic polynomial in n. If the rank of A is r, 1 ≤ r ≤ n − 1, the number of singular factors in this factorization is n − r and this number is minimal. Specializing to the stochastic case, this means that every stochastic matrix A of order n ≥ 2 is a product of at most q(n) elementary stochastic and inverse elementary stochastic matrices, plus a transposition matrix if det A < 0, etc. This factorization can be refined so that for any given ε > 0 no elementary factor has a negative (off-diagonal) entry exceeding ε in absolute value.