×

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.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A23 Factorization of matrices
15B10 Orthogonal matrices
15B35 Sign pattern matrices
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Brualdi, R.A.; Shader, B.L., Matrices of sign-solvable linear systems, (1995), Cambridge University Press Cambridge · Zbl 0833.15002
[2] Eschenbach, C.A.; Hall, F.J.; Harrell, D.L.; Li, Zhongshan, When does the inverse have the same sign pattern as the transpose?, Czechoslovak math. J., 49, 255-275, (1999) · Zbl 0954.15013
[3] Fiedler, M., Relations between the diagonal elements of two mutually inverse positive definite matrices, Czechoslovak math. J., 14, 39-51, (1964) · Zbl 0158.28005
[4] Fiedler, M., Complementary basic matrices, Linear algebra appl., 384, 199-206, (2004) · Zbl 1059.15026
[5] Fiedler, M., Intrinsic products and factorizations, Linear algebra appl., 428, 5-13, (2008) · Zbl 1135.15009
[6] Fiedler, M., Notes on Hilbert and Cauchy matrices, Linear algebra appl., 432, 351-356, (2010) · Zbl 1209.15029
[7] Fiedler, M.; Hall, F.J., Some inheritance properties for complementary basic matrices, Linear algebra appl., 433, 2060-2069, (2010) · Zbl 1254.15020
[8] Hall, F.J.; Li, Zhongshan, Sign pattern matrices, (), (Chapter 33) · Zbl 0997.15010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.