缺陷指數為1的矩陣之研究; A Study on Matrices of Defect Index One

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/7937

Title:	缺陷指數為1的矩陣之研究;A Study on Matrices of Defect Index One
Authors:	吳思潔;Szu-Chieh Wu
Contributors:	數學研究所
Keywords:	數值域;缺陷指數;極分解;polar decomposition;numerical range;defect index
Date:	2008-06-13
Issue Date:	2009-09-22 11:09:34 (UTC+8)
Publisher:	國立中央大學圖書館
Abstract:	ㄧ個n階矩陣A其缺陷指數為$rank(I_n-A^ast A)$。本論文探討關於「缺陷指數為1的矩陣」其性質之刻劃。令$mathcal{S}_n ≡ {A in M_n: rank(I_n-A^ast A)=1 and \|lambda\|<1 for all lambda in sigma(A)}$和 $mathcal{S}_n^{-1} ≡ {A in M_n: rank(I_n-A^ast A)=1 and \|lambda\|>1 for all lambda in sigma(A)}$。首先我們發現這兩類矩陣皆為具有缺陷指數為1之基本矩陣，進一步而言，我們證明一矩陣其缺陷指數為1之充分必要條件為它可分解成一個么正矩陣和一個$mathcal{S}_n$ 矩陣的直和或是一個么正矩陣和一個$mathcal{S}_n^{-1}$矩陣的直和。此外，我們也針對$mathcal{S}_n^{-1}$矩陣給一個完整的刻劃以及它們的極分解。亦證明每一個$mathcal{S}_n^{-1}$ 矩陣均具有循環、不可分解、且其數值域之邊界為一代數曲線。並給出$mathcal{S}_n^{-1}$ 矩陣的範數其由特徵值所表示。 Given an $n$-by-$n$ matrix $A$, the dimension of $ran(I_n-A^ast A)$ is called the defect index of $A$. In this thesis, we make a detailed study of matrices $A$ with the property $rank(I_n-A^ast A)=1$. Let $mathcal{S}_n equiv {A in M_n: rank(I_n-A^ast A)=1: and \|lambda\|<1 for all lambda in sigma(A)}$ and $mathcal{S}_n^{-1} equiv {A in M_n: rank(I_n-A^ast A)=1 and \|lambda\|>1 for all lambda in sigma(A)}$. Firstly, we give a complete characterization for matrices of defect index one, namely, $rank(I_n-A^ast A)=1$ if and only if $A$ is unitarily equivalent to either $U oplus B$ or $U oplus C$, where $U$ is a $k times k$ unitary matrix, $0 leq k< n$, $B in mathcal{S}_{n-k}^{-1}$ and $C in mathcal{S}_{n-k}$. Moreover, we also give a complete characterization of $mathcal{S}_n^{-1}$-matrices. We find the polar decompositions of $mathcal{S}_n^{-1}$-matrices. Next, we prove that every $mathcal{S}_n^{-1}$-matrix is irreducible, cyclic, and the boundary of its numerical range is an algebraic curve. Finally, we give the norm of $mathcal{S}_n^{-1}$-matrices in terms of its eigenvalues.
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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