G-matrices, $J$-orthogonal matrices, and their sign patterns

Frank J. Hall; Miroslav Rozložník

Czechoslovak Mathematical Journal (2016)

• Volume: 66, Issue: 3, page 653-670
• ISSN: 0011-4642

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Abstract

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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 $A^{-\mathrm{T}}=D_{1}A{D}_2$, where $A^{-\mathrm{T}}$ denotes the transpose of the inverse of $A$. Denote by $J=\mathrm{diag}(\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^{\mathrm{T}}JQ=J$. 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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Hall, Frank J., and Rozložník, Miroslav. "G-matrices, $J$-orthogonal matrices, and their sign patterns." Czechoslovak Mathematical Journal 66.3 (2016): 653-670. <http://eudml.org/doc/286845>.

@article{Hall2016,
abstract = {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 $Q^\{\rm T\}J Q=J$. 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.},
author = {Hall, Frank J., Rozložník, Miroslav},
journal = {Czechoslovak Mathematical Journal},
keywords = {G-matrix; $J$-orthogonal matrix; Cauchy matrix; sign pattern matrix},
language = {eng},
number = {3},
pages = {653-670},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {G-matrices, $J$-orthogonal matrices, and their sign patterns},
url = {http://eudml.org/doc/286845},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Hall, Frank J.
AU - Rozložník, Miroslav
TI - G-matrices, $J$-orthogonal matrices, and their sign patterns
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 653
EP - 670
AB - 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 $Q^{\rm T}J Q=J$. 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.
LA - eng
KW - G-matrix; $J$-orthogonal matrix; Cauchy matrix; sign pattern matrix
UR - http://eudml.org/doc/286845
ER -

References

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