Principal submatrices of co-order one with the biggest Perron root Original Research Article

Let A be an irreducible nonnegative matrix, w be any of its indices, and A−w be the principal submatrix of co-order one obtained from A by deleting the wth column and row. Denote by Vext(A) the set of indices w such that A−w has the biggest Perron root (among all the principal submatrices of co-order one of the original matrix A). We prove that exactly one Jordan block corresponds to the Perron root λ(A−w) of A−w for every wset membership, variantVext(A). If its size is strictly greater than one for some wset membership, variantVext(A), then the original matrix A is permutationally similar to a lower Hessenberg matrix with positive entries on the superdiagonal and in the left lower corner (in other words, the digraph D(A) of A has a Hamiltonian circuit and its diameter is one less than its order). In the opposite case for any wset membership, variantVext(A), there is a unique path γ={wi}i=0p going through w in D(A) such that(1) A−wi has the biggest Perron root for i=0,…,p;(2) A−w0 has a right positive Perron eigenvector;(3) A−wp has a left positive Perron eigenvector;(4) A−wi has neither a left nor a right positive Perron eigenvector for i=1,…,p−1.