On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matrices

Abstract

Let A be an irreducible nonnegative matrix and λ( A) be the Perron root (spectral radius) of A. Denote by λ min( A) the minimum of the Perron roots of all the principal submatrices of co-order one. It is well known that the interval ( λ min( A), λ( A)) does not contain any eigenvalues of A. Consider any principal submatrix A− v of co-order one whose Perron root is equal to λ min( A). We show that the Jordan structure of λ min( A) as an eigenvalue of A is obtained from that of the Perron root of A− v as follows: one largest Jordan block disappears and the others remain the same. So, if only one Jordan block corresponds to the Perron root of the submatrix, then λ min( A) is not an eigenvalue of A. By Schneider’s theorem, this holds if and only if there is a Hamiltonian chain in the singular digraph of A− v. In the general case the Jordan structure for the Perron root of the submatrix A− v and therefore that for the eigenvalue λ min( A) of A can be arbitrary. But if the Perron root λ( A− w) of a principal submatrix A− w of co-order one is strictly greater than λ min( A), then λ( A− w) is a simple eigenvalue of A− w. We also obtain different representations for the generalized eigenvectors corresponding to the eigenvalues of A contained in the annulus { λ: λ min( A)<| λ|< λ( A)}.