The structure of Schur complements in hollow, symmetric nonnegative matrices with two nonpositive eigenvalues

Abstract

The Schur complement structure with respect to principal submatrices, in hollow, symmetric nonnegative matrices is investigated, with an emphasis on such matrices that have only two nonpositive eigenvalues. It is shown that a wide family of such Schur complements simply follows a unique and surprising structure that can be fully described in a graph theoretical language, and is predictable from the entries. For larger numbers of nonpositive eigenvalues, conjectures regarding connections to polyhedra are also presented, and proved in a special case. Relations to copositive matrices and Morishima matrices are described as well.

Keywords:

• copositive matrices
• cycle graphs
• Morishima matrices
• nonpositive eigenvalues
• Schur complements

AMS Subject Classifications:

• 15A18
• 15B48
• 15B57
• 15A42