On linear preservers of permanental rank

Abstract

Let ${\mathrm{Mat}}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank $\mathrm{prk}(A)$ of a matrix $A\in {\mathrm{Mat}}_n(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By ${\Lambda }^k$ and ${\Lambda }^{\le k}$ we denote the subsets of matrices $A\in {\mathrm{Mat}}_n(\mathbb{F})$ with $\mathrm{prk}(A)=k$ and $\mathrm{prk}(A)\le k$, respectively. In this paper for each $1\le k\le n-1$ we obtain a complete characterization of linear maps $T:{\mathrm{Mat}}_n(\mathbb{F})\to {\mathrm{Mat}}_n(\mathbb{F})$ satisfying $T({\Lambda }^{\le k})={\Lambda }^{\le k}$ or bijective linear maps satisfying $T({\Lambda }^{\le k})\subseteq {\Lambda }^{\le k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then ${\Lambda }^k$ is Zariski dense in ${\Lambda }^{\le k}$ and apply this to describe such bijective linear maps satisfying $T({\Lambda }^k)\subseteq {\Lambda }^k$.

15 pages, minor corrections

References in corpus (1)

• Maximal Generalized Rank in Graphical Matrix Spaces