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 $A^-T= D_1 AD_2$, where $A^-T$ denotes the transpose of the inverse of $A$. Denote by $J = diag(pm1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^ TJ Q=obreak J$. 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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机译：如果$ A $是非奇数并且存在非奇数对角矩阵$ D_1 $和$ D_2 $，使得$ A ^ -T = D_1 AD_2 $，则真实矩阵$ A $是G矩阵，其中$ A ^ -T $表示$ A $倒数的转置。用$ J = diag（ pm1）$表示一个对角（签名）矩阵，每个对角矩阵为$ + 1 $或$ -1 $。如果$ Q ^ TJ Q = nobreak J $，则非奇异实数矩阵$ Q $称为$ J $-正交的。这些矩阵之间建立了许多连接。特别是，当且仅当$ A $对角线（具有正对角线）等效于$ J $正交矩阵的列置换时，矩阵$ A $才是G矩阵。开始研究$ J $正交矩阵的符号模式。可以看出，G矩阵的符号模式恰好是$ J $正交矩阵的符号模式的列排列。展示了某些有趣的$ J $正交矩阵构造。结果表明，每个对称楼梯符号图案矩阵都允许一个$ J $正交矩阵。还考虑了可能的符号$ J $正交条件。提供了一些示例和未解决的问题。
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