臺灣博碩士論文加值系統

冪零--中心化子方法是一種能通過一個具有非對稱強譜性質的冪零矩陣來構造譜自由符號矩陣的方法。因此，使用這種方法的關鍵是找到一個具有非對稱強譜性質的冪零矩陣。但找到這樣的矩陣並非易事。在這篇論文中，我們參考了對稱版本的矩陣解放引理並提出了非對稱版本的矩陣解放引理，這使得我們可以通過一個不具有非對稱強譜性質的冪零矩陣來得到一個具有非對稱強譜性質的冪零矩陣。當我們將矩陣解放引理與冪零--中心化子方法結合起來，我們便可用相對一般的冪零矩陣去構造譜自由符號矩陣。

The nilpotent-centralizer method is a way to construct spectrally arbitrary pattern (SAP) through a nilpotent matrix with the non-symmetric strong spectral property (nSSP). The key to using this method is to find a nilpotent matrix with nSSP, but it is not easy to find such a matrix. In this thesis, we refer to the symmetric version of the Matrix Liberation Lemma and propose the non-symmetric version of the Matrix Liberation Lemma. This allows us to obtain a nilpotent matrix with nSSP through a nilpotent matrix without nSSP. Combining the Matrix Liberation Lemma and the nilpotent-centralizer method, we can use relatively general nilpotent matrices to construct spectrally arbitrary patterns.

論文審定書 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Spectrally arbitrary pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 The nilpotent-Jacobian method and the nilpotent-centralizer method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 The nilpotent-Jacobian method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 The nilpotent-centralizer method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 The Matrix Liberation Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1 Non-symmetric strong spectral property . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Tangent spaces and normal spaces of pertinent manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 The Matrix Liberation Lemma . . . . . . . . . . . 11
5 Constructing SAPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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