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

Abstract

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 V ext( 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∈ V ext( A). If its size is strictly greater than one for some w∈ V ext( 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∈ V ext( A), there is a unique path γ={ w i } i=0 p going through w in D( A) such that (1) A− w i has the biggest Perron root for i=0,…, p; (2) A− w 0 has a right positive Perron eigenvector; (3) A− w p has a left positive Perron eigenvector; (4) A− w i 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− w 0,…, A− w p combined inherit the property of irreducibility. We also show that A− w is irreducible for every w∈ V ext( 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∈ V ext( A) or there exist exactly two indices w in and w out in V ext( A) such that A− w in and A− w out 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.