论文标题
有限维空间中线性关系的Jordan样分解
A Jordan-like decomposition for linear relations in finite-dimensional spaces
论文作者
论文摘要
Square Matrix $ a $具有通常的Jordan规范表格，该表格描述了$ a $的结构，并通过特征值和相应的Jordan块。如果$ a $是有限维线性空间$ {\ mathfrak h} $中的线性关系（即，$ a $是$ {\ mathfrak h} \ times {\ mathfrak h} $的线性子空间，则可以被视为多价线性操作员），那么），则是一个富有的结构。除经典的约旦连锁店（在笛卡尔产品中解释$ {\ mathfrak h} \ times {\ mathfrak h} $）外，还有三个类别的链条：从零开始的链条（eigenvalue Infinity的链条），链条的链条，链条从零开始，并在零链中（以及独立的独立链）（和单独的链）（和单独的链），以及链），链），以及链），链），链），链），链），链条），以及链）。这四种类型的链条产生了线性关系$ a $的直接总和分解（类似Jordan的分解）。在这种分解中，有一个完全奇异的部分，其延伸的复合面是特征值。通常的约旦部分，与有限的适当特征值相对应；与特征值$ \ infty $相对应的约旦部分；以及一个多派式的，即根本没有特征值的部分。此外，类似约旦的分解表现出一定的独特性，缩小了早期结果的差距。呈现纯粹是代数，仅使用线性空间的结构。此外，演示文稿具有统一的特征：上述每种类型都是通过适当选择的商空间序列构建的。空间的尺寸是Weyr特性，它们独特地决定了线性关系的Jordan样分解。
A square matrix $A$ has the usual Jordan canonical form that describes the structure of $A$ via eigenvalues and the corresponding Jordan blocks. If $A$ is a linear relation in a finite-dimensional linear space ${\mathfrak H}$ (i.e., $A$ is a linear subspace of ${\mathfrak H} \times {\mathfrak H}$ and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product ${\mathfrak H} \times {\mathfrak H}$), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation $A$. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue $\infty$; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation.
