Two n × n G-classes of matrices having finite intersection

Abstract

Abstract Let M n {{\bf{M}}}_{n} be the set of all n × n n\times n real matrices. A nonsingular matrix A ∈ M n A\in {{\bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} such that A − T = D 1 A D 2 {A}^{-T}={D}_{1}A{D}_{2} . For fixed nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} , let G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } , {\mathbb{G}}\left({D}_{1},{D}_{2})=\left\{A\in {{\bf{M}}}_{n}:{A}^{-T}={D}_{1}A{D}_{2}\right\}, which is called a G-class. The purpose of this short article is to answer the following open question in the affirmative: do there exist two n × n n\times n G-classes having finite intersection when n ≥ 3 n\ge 3 ?