On the bounds for Perron roots of the arithmetic symmetrization of nonnegative matrices

An nxn matrix A = [aij] is symmetric if aij = aji, and A is nonnegative (A ³ 0) ifaij ³ 0 for 1 £ i, j £ n. For a nonsymmetric matrix A, the arithmetic symmetrization M(A) is defined by M(A) = [mij]. Where mij = 1/2 (aij + aji). The Perron root r(A) of a nonnegative A is a nonnegative eigenvalue of A such that r(A ) ³|l| for alleigen values of A. This article is devoted to determining a sequence of lower and upper bounds for rM(A).