Two n × n G-classes of matrices having finite intersection - Golshan Setareh; Armandnejad Ali; Hall Frank J. (Special Matrices, 2022)
Let Mn{{\bf{M}}}_{n} be the set of all n×nn× n real matrices. A nonsingular matrix A∈MnA\in {{\bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D1D_{1} and D2D_{2} such that A−T=D1AD2A^{-T}=D_{1}AD_{2}. For fixed nonsingular diagonal matrices D1D_{1} and D2D_{2}, let G(D1,D2)={A∈Mn:A−T=D1AD2},{\mathbb{G}}\left(D_{1},D_{2})=\left\{A\in {{\bf{M}}}_{n}:A^{-T}=D_{1}AD_{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×nn× n G-classes having finite intersection when n≥3n\ge 3?