两个矩阵集合的不可约准则
Irreducibility criterion for the set of two matrices
摘要：
利用与矩阵$A_n$的“支撑”相关的子代数给出了两个$n\times n$矩阵$Λ_n$和$A_n$的集合的不可约性、Schur不可约性和不可分解性的判别准则，其中$Λ_n$是具有不同非零特征值的对角矩阵，$A_n$是任意对角矩阵.给出了代数${\rm Mat}(n,{\mathbb C})$的所有极大子代数的列表以及与这两个矩阵相联系的相应的不变子空间的列表。相应子代数的性质用与第二矩阵的支集相关的图来表示。对于任意的$n$，我们描述生成代数${\rm Mat}(n,{\mathbb C})$的初等矩阵$E_{km}$的所有极小子集。
英文摘要：
We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two $n\times n$ matrices $Λ_n$ and $A_n$ in terms of the subalgebra associated with the "support" of the matrix $A_n$, where $Λ_n$ is a diagonal matrix with different non zeros eigenvalues and $A_n$ is an arbitrary one. The list of all maximal subalgebras of the algebra ${\rm Mat}(n,{\mathbb C})$ and the list of the corresponding invariant subspaces connected with these two matrices is also given. The properties of the corresponding subalgebras are expressed in terms of the graphs associated with the support of the second matrix. For arbitrary $n$ we describe all minimal subsets of the elementary matrices $E_{km}$ that generate the algebra ${\rm Mat}(n,{\mathbb C})$.
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