Schur complements and its applications to symmetric nonnegative and Z-matrices

Abstract

In [Linear Algebra Appl. 177 (1992) 137] Smith proved that if H is a Hermitian semidefinite matrix and A is a nonsingular principal submatrix, then the eigenvalues of the Schur complement H/ A interlace those of H. In this paper, we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve Perron complements. For an irreducible symmetric nonnegative matrix, we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order. We apply our results to an irreducible symmetric Z-matrix and to the adjacency matrix or the general Laplacian matrix of a connected weighted graph. The equality cases for the bounds for spectral radii or least eigenvalues are also examined.