An elementary proof of Chollet’s permanent conjecture for 4 × 4 real matrices

George Hutchinson

doi:10.1515/spma-2020-0126

Abstract

Abstract A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.

Highlights

IntroductionIn 1982 [1], John Chollet introduced a conjecture concerning the permanent of positive semide nite (psd) matrices
In 1982 [1], John Chollet introduced a conjecture concerning the permanent of positive semide nite matrices. Before we state this conjecture, we remind the reader of the de nition of the matrix permanent: De nition 1.1
The purpose of this paper is to provide a proof that Conjecture 1.2 is true for × real matrices

Summary

Introduction

In 1982 [1], John Chollet introduced a conjecture concerning the permanent of positive semide nite (psd) matrices. Before we state this conjecture, we remind the reader of the de nition of the matrix permanent: De nition 1.1. Let A = (aij) be an n × n matrix with complex entries, and let Sn denote the symmetric group on n elements. The permanent of A, denoted per(A), is the complex number n per(A) = ai,σ(i) . In its original form, Chollet’s conjecture was given as: Conjecture 1.2 ([1]). Let A, B be two n × n positive semide nite matrices.

Objectives

Conclusion