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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.19594 |
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| _version_ | 1866914989114130432 |
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| author | Boussaïri, Abderrahim Chergui, Brahim Sarir, Zaineb Zouagui, Mohamed |
| author_facet | Boussaïri, Abderrahim Chergui, Brahim Sarir, Zaineb Zouagui, Mohamed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19594 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasi-orthogonal extension of skew-symmetric matrices Boussaïri, Abderrahim Chergui, Brahim Sarir, Zaineb Zouagui, Mohamed Combinatorics 15A18, 15B10 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. |
| title | Quasi-orthogonal extension of skew-symmetric matrices |
| topic | Combinatorics 15A18, 15B10 |
| url | https://arxiv.org/abs/2410.19594 |