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

Let A   be an n×nn×n nonnegative irreducible matrix, let A[α]A[α] be the principal submatrix of AA based on the nonempty ordered subset αα of {1,2,…,n}{1,2,…,n}, and define the generalized Perron complement of A[α]A[α] by Pt(A/A[α])Pt(A/A[α]), i.e., Pt(A/A[α])=A[β]+A[β,α](tI-A[α])-1A[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 t0t0 and the maximum of the row sums of Pt0(A/A[α])Pt0(A/A[α]), synchronously, a lower bound on Perron root is expressed by the minimum of the given parameter t0t0 and the minimum of the row sums of Pt0(A/A[α])Pt0(A/A[α]). It is also shown how to choose the parameter tt after αα to get tighter upper and lower bounds of ρ(A)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.