论文标题
列举相似性类别的部分线性变换
Enumerating partial linear transformations in a similarity class
论文作者
论文摘要
令$ v $为有限维矢量空间$ {\ mathbb f} _q $，假设$ w $和$ \ widetilde {w} $是$ v $的子空间。两个线性转换$ t：w \ to v $和$ \ widetilde {t}：\ widetilde {w} \ v $，如果存在线性同构$ s：v \ t $ s with $ sw = \ sw = \ wideTilde {w} $ s $ s $ s \ cool con $ cool t = t =鉴于在子空间$ w $ $ v $上定义的线性地图$ t $，我们给出了类似于$ t $的线性地图数量的明确公式。我们的结果扩展了菲利普大厅的定理，该定理解决了$ w = v $的情况，其中上述问题等同于在共轭类中计算$ {\ mathbb f} _q $以上的平方矩阵数量。
Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\widetilde{W}$ such that $S\circ T=\widetilde{T}\circ S $. Given a linear map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the number of linear maps that are similar to $T$. Our results extend a theorem of Philip Hall that settles the case $W=V$ where the above problem is equivalent to counting the number of square matrices over ${\mathbb F}_q$ in a conjugacy class.
