Topological properties of J-orthogonal matrices, part II

Abstract

This paper is a continuation of the article ‘Topological properties of J-orthogonal matrices’, Linear and Multilinear Algebra 66(2018), 2524–2533, by the authors. Let ${\mathbf{M}}_n$ be the set of all $n\times n$ real matrices. A matrix $J\in {\mathbf{M}}_n$ is said to be a signature matrix if J is diagonal and its diagonal entries are $\pm 1$. If J is a signature matrix, a nonsingular matrix $A\in {\mathbf{M}}_n$ is said to be a J-orthogonal matrix if $A^{\mathrm{\top }}JA=J$. Let ${\mathrm{\Omega }}_n$ be the set of all $n\times n$, J-orthogonal matrices. In this paper some further interesting properties of these matrices are obtained. In particular, an open question stated in the preceding article about ${\mathrm{\Omega }}_n$ is answered. Proposition 3.2 on the characterization of J-orthogonal matrices in the paper ‘J-orthogonal matrices: properties and generation’, SIAM Review 45(2003), 504–519, by N. J. Higham is again heavily used. The standard linear operators $T:{\mathbf{M}}_n\to {\mathbf{M}}_n$ that strongly preserve J-orthogonal matrices, i.e. $T\left(A\right)$ is J-orthogonal if and only if A is J-orthogonal are characterized.

COMMUNICATED BY:

• J. F. Queir$\acute{\mathrm{o}}$

Keywords:

• Signature matrix
• signed permutation matrix
• linear preservers
• j-orthogonal matrix

2010 Mathematics Subject Classifications:

• Primary: 15A04
• Secondary: 15A21

Acknowledgments

The authors thank Professor Michael Stewart of Georgia State University for the elegant proof of Lemma 2.8. Also they thank the referee for very good suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.