Resolvent comparison for Hermitian block matrices and Kelmans transformations

Abstract

A Kelmans transformation increases or preserves the largest eigenvalue of a symmetric nonnegative matrix. We determine when this comparison holds for Hermitian block extensions with a fixed coupling and an arbitrary common complementary block. An orthogonal change of basis and Schur complementation give an exact formula for the compressed resolvent difference as a difference of two rank-one matrices. This formula characterizes admissible coupling subspaces by proportionality of projected resolvent vectors. Polynomial coefficient kernels describe all such subspaces, and cyclic subspaces give dimension bounds and a maximum-dimension result for a two-block family. We also prove a general block comparison theorem and give monotonicity and equality criteria for nonnegative rank-one couplings. Applications concern graph joins, complex attachments, and Hermitian adjacency matrices.