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

Let A be an irreducible nonnegative matrix and λ(A) be the Perron root (spectral radius) of A. Denote by λmin(A) the minimum of the Perron roots of all the principal submatrices of co-order one. It is well known that the interval (λmin(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 λmin(A). We show that the Jordan structure of λmin(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 λmin(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 λmin(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 λmin(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:λmin(A)<λ<λ(A)}.