A NOTE ON THE EIGENVALUES OF A PRIMITIVE MATRIX WITH LARGE EXPONENT

Let A be a primitive stochastic matrix of order n greater than or equa l to 7 and exponent at least [[(n - 1)(2) + 1]/2] + 2. We describe the general form of the characteristic polynomial of A, and prove that A must have at least 2[(n - 4)/4] complex eigenvalues of modulus at grea ter than {1/2 sin[pi/n-1]}(2/(n-1)) (observe that this last quantity t ends to 1 as n --> infinity). Both combinatorial and algebraic argumen ts are used to establish the result. (C) Elsevier Science Inc., 1997.