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

Abstract

Let Mn{{\bf{M}}}_{n} be the set of all n×nn\times n real matrices. A nonsingular matrix A∈MnA\in {{\bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D1{D}_{1} and D2{D}_{2} such that A−T=D1AD2{A}^{-T}={D}_{1}A{D}_{2}. For fixed nonsingular diagonal matrices D1{D}_{1} and D2{D}_{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}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×nn\times n G-classes having finite intersection when n≥3n\ge 3?

Keywords

• g-matrix
• g-class
• inertia of matrices
• primary: 15b10
• secondary: 15a30