矩陣束之幾何結構

Abstract

本論文研究非正規矩陣束$(E,A)$的幾何結構以及相應的自治離散時間描述子系統$Ex_{i+1}=Ax_i$。我們展示了Wong氏序列 [Wong, 1974] 的維度結構與Kronecker典型形式之間的聯繫。我們證明了狀態空間可以分解為奇異子空間、非平凡譜子空間和multishift子空間的直和。我們進一步根據相應的商空間結構找到各個子空間的Kronecker基底。此外，我們證明了各個子空間的Kronecker基底恰由奇異鏈、Jordan鏈和multishift鏈構成。這為非正規矩陣束的Kronecker典型形式提供了一種新的幾何證明方法，並且提供更多的幾何直觀。最後，離散時間描述子系統$Ex_{i+1}=Ax_i$的解行為將被完整刻劃。我們也給出數值範例用於說明。

Geometric structure of non-regular matrix pencils $(E,A)$ and the corresponding autonomous discrete-time descriptor systems $Ex_{i+1}=Ax_i$ are studied. We show the connection between the dimension structure of Wong's sequences [Wong, 1974] and the Kronecker canonical form. We also show that the state space can be decomposed as the sum of singularity subspace, nontrivial spectral subspace and the multishift subspace. We further find the Kronecker basis for each subspace in terms of pertinent quotient spaces. Moreover, we prove that the Kronecker basis for each subspace consists of singular chains, Jordan chains and the multishift chains. This provides a new geometric approach to the Kronecker canonical form of a non-regular matrix pencil, and gives more geometric insight. Finally, the solution behavior of the discrete-time descriptor systems $Ex_{i+1}=Ax_i$ is given. Some numerical examples are given for illustration.