 English  |  正體中文  |  简体中文  |  Items with full text/Total items : 75533/75533 (100%) Visitors : 27366397      Online Users : 431
 Scope All of NCUIR 理學院    數學研究所       --博碩士論文 Tips: please add "double quotation mark" for query phrases to get precise resultsplease goto advance search for comprehansive author search Adv. Search
 Home ‧ Login ‧ Upload ‧ Help ‧ About ‧ Administer NCU Institutional Repository > 理學院 > 數學研究所 > 博碩士論文 >  Item 987654321/7937

 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: [數學研究所] 博碩士論文

Files in This Item:

File SizeFormat