Affine subspace of matrices with constant rank
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866929609000353792 |
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| author | Rubei, Elena |
| author_facet | Rubei, Elena |
| contents | For every $m,n \in \mathbb{N}$ and every field $K$, let $M(m \times n, K)$ be the vector space of the $(m \times n)$-matrices over $K$ and let $S(n,K)$ be the vector space of the symmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $M(m \times n, K)$ or of $S(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}^K(m \times n; r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $M(m \times n, K)$ of constant rank } r\}$$ $${\cal A}_{sym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $S(n,K)$ of constant rank } r\}$$ $$a^K(m \times n;r) = \max \{\dim S \mid S \in {\cal A}^K(m \times n; r ) \}.$$ $$a_{sym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{sym}^K(n,r) \}.$$
In this paper we prove the following two formulas for $r \leq m \leq n$: $$a_{sym}^{\mathbb{R}}(n;r) \leq \left\lfloor \frac{r}{2} \right\rfloor \left(n- \left\lfloor \frac{r}{2} \right\rfloor\right)$$ $$a^{\mathbb{R}}(m \times n;r) = r(n-r)+ \frac{r(r-1)}{2} .$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14997 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Affine subspace of matrices with constant rank Rubei, Elena Rings and Algebras 15A30 For every $m,n \in \mathbb{N}$ and every field $K$, let $M(m \times n, K)$ be the vector space of the $(m \times n)$-matrices over $K$ and let $S(n,K)$ be the vector space of the symmetric $(n \times n)$-matrices over $K$. We say that an affine subspace $S$ of $M(m \times n, K)$ or of $S(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define $${\cal A}^K(m \times n; r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $M(m \times n, K)$ of constant rank } r\}$$ $${\cal A}_{sym}^K(n;r)= \{ S \;| \; S \; \mbox{\rm affine subsapce of $S(n,K)$ of constant rank } r\}$$ $$a^K(m \times n;r) = \max \{\dim S \mid S \in {\cal A}^K(m \times n; r ) \}.$$ $$a_{sym}^K(n;r) = \max \{\dim S \mid S \in {\cal A}_{sym}^K(n,r) \}.$$ In this paper we prove the following two formulas for $r \leq m \leq n$: $$a_{sym}^{\mathbb{R}}(n;r) \leq \left\lfloor \frac{r}{2} \right\rfloor \left(n- \left\lfloor \frac{r}{2} \right\rfloor\right)$$ $$a^{\mathbb{R}}(m \times n;r) = r(n-r)+ \frac{r(r-1)}{2} .$$ |
| title | Affine subspace of matrices with constant rank |
| topic | Rings and Algebras 15A30 |
| url | https://arxiv.org/abs/2111.14997 |