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arXiv · 2410.19594

Quasi-orthogonal extension of skew-symmetric matrices

Abderrahim Boussaïri·Brahim Chergui·Zaineb Sarir·Mohamed Zouagui
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Abstract

A real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI$, for some positive real number $q$. We prove that any $n\times n$ skew-symmetric matrix $S$ is a principal sub-matrix of a skew-symmetric quasi-orthogonal matrix $Q$, called a quasi-orthogonal extension of $S$. Moreover, we determine the least integer $d$ such that $S$ has a quasi-orthogonal extension of order $n+d$. This integer is called the quasi-orthogonality index of $S$. Lastly, we give a spectral characterization of skew-adjacency matrices of tournaments with quasi-orthogonality index at most three.

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Abderrahim Boussaïri, Brahim Chergui, Zaineb Sarir, Mohamed Zouagui. 2024-10-25. Quasi-orthogonal extension of skew-symmetric matrices. https://arxiv.org/abs/2410.19594

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