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

AbstractLet 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 w∈Vext(A). If its size is strictly greater than one for some w∈Vext(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 w∈Vext(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.Thus, by the spectral criterion for a nonnegative matrix to be irreducible, the submatrices A−w0,…,A−wp combined inherit the property of irreducibility. We also show that A−w is irreducible for every w∈Vext(A) if any of the following holds:(1)A is symmetric;(2)every column (row) of A has at least two positive nondiagonal entries;(3)A has at least two columns (rows) all of whose entries are positive.If A is an irreducible tournament matrix, then either A−w is also irreducible for any w∈Vext(A) or there exist exactly two indices win and wout in Vext(A) such that A−win and A−wout are reducible. In the last case any other principal submatrix of co-order one is irreducible. This shows that in the general case, a one-vertex-deleted subdigraph with the biggest Perron root need not have the best connectivity properties among all one-vertex-deleted subdigraphs of a given strongly connected digraph D