AN APPROACH TO THE PERMANENTAL-DOMINANCE CONJECTURE

The permanental-dominance conjecture for positive semidefinite Hermitian matrices A has attracted much interest in the last ten years. A stronger conjecture, that the maximal eigenvalue of the Schur power matrix PI(A), is per A, would if true imply the dominance of the permanent. We prove that should the maximum-eigenvalue conjecture fail in the real case, then the smallest n for which it fails must be such that it fails at a singular matrix having certain properties, including zero row sums. Many of our results were also obtained independently by J. P. Holmes and Tom Pate at Auburn University.