论文标题
保留线性变换不变子空间的晶格
Collineations preserving the lattice of invariant subspaces of a linear transformation
论文作者
论文摘要
给定有限维的复杂矢量空间$ \ ev $的线性转换$ a $，在本文中，我们研究了$ \ col（a）$由$ \ ev $上的那些可逆线性变换$ s $组成的$ \ ev $的映射为$ \ v $，该$ \φ_s$定义为$φ_s\ colon \ em \ em \ em \ em \ em \ em em y em em em em em em em as as thaud as thaut last rath auts thaut（em em） $ a $的所有不变子空间。通过使用$ a $的主要分解，我们首先将表征$ \ col（a）$表征的问题描述为特征给定的nilpotent线性变换$ n $的组$ \ col（n）$的问题。尽管$ \ col（n）$总是包含$ n $的通勤$（n）'$的所有可逆线性转换，但它始终包含在反射盖$ \ alg \ alg \ lat（n）'$（n）'$中。我们证明$ \ col（n）$是$（\ alg \ lat（n）'）^{ - 1} $的适当子组，并且仅当$ n $的约旦分解中至少有两个jordan块为$ 2 $或更多。我们还确定组$ \ col（\ bdj_2 \ oplus \ bdj_2）$。
Given a linear transformation $A$ on a finite-dimensional complex vector space $\eV$, in this paper we study the group $\Col(A)$ consisting of those invertible linear transformations $S$ on $\eV$ for which the mapping $Φ_S$ defined as $Φ_S\colon \eM\mapsto S\eM$ is an automorphism of the lattice $\Lat(A)$ of all invariant subspaces of $A$. By using the primary decomposition of $A$, we first reduce the problem of characterizing $\Col(A)$ to the problem of characterizing the group $\Col(N)$ of a given nilpotent linear transformation $N$. While $\Col(N)$ always contains all invertible linear transformations of the commutant $(N)'$ of $N$, it is always contained in the reflexive cover $\Alg\Lat(N)'$ of $(N)'$. We prove that $\Col(N)$ is a proper subgroup of $(\Alg\Lat(N)')^{-1}$ if and only if at least two Jordan blocks in the Jordan decomposition of $N$ are of dimension $2$ or more. We also determine the group $\Col(\bdJ_2\oplus \bdJ_2)$.
