The lower and upper bounds on Perron root of nonnegative irreducible matrices

Let A be an n × n nonnegative irreducible matrix, let A [ α ] be the principal submatrix of A based on the nonempty ordered subset α of { 1 , 2 , … , n } , and define the generalized Perron complement of A [ α ] by P t ( A / A [ α ] ) , i.e., P t ( A / A [ α ] ) = A [ β ] + A [ β , α ] ( tI - A [ α ] ) - 1 A [ α , β ] , t > ρ ( A [ α ] ) . This paper gives the upper and lower bounds on the Perron root of A. An upper bound on Perron root is derived from the maximum of the given parameter t 0 and the maximum of the row sums of P t 0 ( A / A [ α ] ) , synchronously, a lower bound on Perron root is expressed by the minimum of the given parameter t 0 and the minimum of the row sums of P t 0 ( A / A [ α ] ) . It is also shown how to choose the parameter t after α to get tighter upper and lower bounds of ρ ( A ) . Several numerical examples are presented to show that our method compared with the methods in [L.Z. Lu, M.K. Ng, Locations of Perron roots, Linear Algebra Appl. 392 (2004) 103–117.] is more effective.