SOME RESULTS ON THE INTERSECTION OF G-CLASSES OF MATRICES

Abstract

‎The rich collection of G-matrices originated in a 2012 paper by Fiedler and Hall‎. ‎Let $\mathbf{M}_n$ be the set of all $n\times n$ real matrices‎. ‎A nonsingular matrix $A\in \mathbf{M}_n$ is called a G-matrix if there exist nonsingular diagonal matrices‎
‎$D_1$ and $D_2$ such that $ A^{- ‎T} = D_1AD_2$‎, ‎where $A^{-T}$ denotes the transpose of the inverse of $A$‎. ‎For fixed nonsingular diagonal matrices‎
‎$D_1$ and $D_2$‎, ‎let $\mathbb{G}(D_{1},D_{2})=\{ A\in \mathbf{M}_n‎: A^{- ‎T} = D_1AD_2 \},$ which is called a G-class‎. ‎In more recent papers‎, ‎G-classes of matrices were studied‎. ‎The purpose of this present work is to find conditions on $ D_{1}$‎, ‎$D_{2}$‎,
‎$D_{3}$ and $D_{4}$ such that the G-classes $ \mathbb{G}(D_{1},D_{2})$ and $\mathbb{G}(D_{3},D_{4}) $ have finite nonempty intersection or empty intersection‎. ‎A main focus of this work is the use of the diagonal matrix‎
‎$D = D_3^{1/2} D_1^{-1/2}$‎. ‎In the case that all the $D_i$ are $n\times n$ diagonal matrices with positive diagonal entries‎,
‎complete characterizations of the G-classes are obtained for the intersection questions‎.