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G-matrix; $J$-orthogonal matrix; Cauchy matrix; sign pattern matrix
A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_1$ and $D_2$ such that $A^{\rm -T}= D_1 AD_2$, where $A^{\rm -T}$ denotes the transpose of the inverse of $A$. Denote by $J = {\rm diag}(\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^{\rm T}J Q=\nobreak J$. Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation of a $J$-orthogonal matrix. An investigation into the sign patterns of the $J$-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the $J$-orthogonal matrices. Some interesting constructions of certain $J$-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a $J$-orthogonal matrix. Sign potentially $J$-orthogonal conditions are also considered. Some examples and open questions are provided.
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