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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/2407.14273 |
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| _version_ | 1866914960488005632 |
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| author | Balasubramanian, Kumar Kaipa, Krishna Khurana, Himanshi |
| author_facet | Balasubramanian, Kumar Kaipa, Krishna Khurana, Himanshi |
| contents | Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $α\in F$ and $A\in \M(n,F)$, let $$Z^α_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=α\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{A,r}^α$. We also solve the generalization of the problem to rectangular matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14273 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the cardinality of matrices with prescribed rank and partial trace over a finite field Balasubramanian, Kumar Kaipa, Krishna Khurana, Himanshi Rings and Algebras 15A03, 15A15 Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $α\in F$ and $A\in \M(n,F)$, let $$Z^α_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=α\right \}.$$ In this article, we solve the problem of determining the cardinality of $Z_{A,r}^α$. We also solve the generalization of the problem to rectangular matrices. |
| title | On the cardinality of matrices with prescribed rank and partial trace over a finite field |
| topic | Rings and Algebras 15A03, 15A15 |
| url | https://arxiv.org/abs/2407.14273 |