All Questions

Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...

Suppose $M_{n}^{k}$ is the number of non-singular $n\times n$ matrices over $\mathbb{F}_2$, that have exactly $k$ non-zero entries. Is there some sort of formula to calculate $M_n^k$? If $k < n$ ...

Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$? If $...

$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$. $B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$. $T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...