A note on the eigenvalues of a primitive matrix with large exponent Original Research Article

Let A be a primitive stochastic matrix of order n greater-or-equal, slanted 7 and exponent at least left floor[(n − 1)2 + 1]/2right floor + 2. We describe the general form of the characteristic polynomial of A, and prove that A must have at least 2left floor(n − 4)/4right floor complex eigenvalues of modulus at greater than image (observe that this last quantity tends to 1 as n → ∞). Both combinatorial and algebraic arguments are used to establish the result.