G-matrices. (English) Zbl 1237.15024

The authors introduce the concept: G-matrix (which is totally different from the concept G-matrix in biology). A nonsingular real square matrix \(A\) is called a G-matrix if there exist nonsingular diagonal matrices \(D_1\) and \(D_2\) such that \((A^{-1})^T= D_1AD_2\). Examples of G-matrices include Cauchy matrices and orthogonal matrices. Although the reason to introduce G-matrices does not seem to be clear, this is an interesting notion in theory. A number of properties of G-matrices are obtained and sign patterns of G-matrices are studied.
By the way, the restrictions on Cauchy matrices in Theorem 2.11 is redundant as mentioned after this theorem. This can easily be seen from the formula on Line 8 of Page 733.


15B05 Toeplitz, Cauchy, and related matrices
15A23 Factorization of matrices
15B10 Orthogonal matrices
15B35 Sign pattern matrices
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