臺灣博碩士論文加值系統

令A是一個n階複數矩陣，A矩陣的數值域可以表示成集合W(A)＝{x*Ax：x屬於Cn,x*x＝1}， 我們要找絕對冪零上三角Toeplitz矩陣的數值域的圓盤性質。假如 A 是絕對冪零上三角Toeplitz矩陣而且它的形式是A(a1,a2,0)屬於M4 或者是A(a1,a2,0,0)屬於M5，我們證明W(A)是一個圓盤而且中心點是原點若且為若a1a2＝0。相同的，假如 A 是絕對冪零上三角Toeplitz矩陣而且它的形式是A(a1,0,a3)屬於M4或者是 A(a1,0,a3,0)屬於M5，我們證明W(A)是一個圓盤而且中心點是原點若且為若a1a3＝0。假如 A 是絕對冪零上三角Toeplitz矩陣而且它的形式是A(a1,0,0,a4)屬於M5 或者是A(a1,0,0,a4,0)屬於M6，我們證明W(A)是一個圓盤而且中心點若且為若a1a4＝0。而且，假如 A 是絕對冪零上三角Toeplitz矩陣而且它的形式是A(a1,0,a3)屬於M4，A(a1,0,0,a4)屬於M5，A(a1,0,…,an-1)屬於Mn ，我們證明W(A)是一個圓盤而且中心點是原點若且為若a1an-1＝0。最後，假如 A 是絕對冪零上三角Toeplitz矩陣而且它的形式是A(0,0,0,…,0,a[n/2],a[n/2]+1,…,an-1 )，我們證明W(A)是一個圓盤而且中心點是原點若且為若a[n/2]a[n/2]+1…an-1=\=0。

Let A be an n-by-n complex matrix. The numerical range of A is theset W(A) = {x*Ax : x belongs to Cn, x*x = 1}. We deal with the circularityproperty of the numerical range of certain nilpotent upper triangularToeplitz matrices. If A is the nilpotent upper triangular Toeplitz matrix of the forms A(a1,a2,0)belongs to M4 or A(a1,a2,0,0)belongs to M5, we provethat W(A) is a circular disk centered at the origin if and only if a1a2=0.Similarly, if A is the nilpotent upper triangular Toeplitz matrix of theforms A(a1,0,a3)belongs to M4 or A(a1,0,a3,0)belongs to M5, W(A) is a circular disk centered at the origin if and only if a1a3=0. If A is the nilpotentupper triangular Toeplitz matrix of the forms A(a1,0,0,a4)belongs to M5 orA(a1,0,0,a4,0)belongs to M6, we prove that W(A) is a circular disk centeredat the origin if and only if a1a4=0. Furthermore, if A is the nilpotent
upper triangular Toeplitz matrix of the forms A(a1,0,a3)belongs to M4 orA(a1,0,0,a4)belongs to M5, A(a1,0, .., 0,an−1)belongs to Mn, we prove that W(A) isa circular disk centered at the origin if and only if a1an−1=0. Finally,if A is the nilpotent upper triangular Toeplitz matrix of the forms
A(0, 0, .., 0, a[n/2], a[n/2]+1, .., an−1)belongs to Mn, we prove that W(A) is a circular
disk centered at the origin if and only if an/2a(n/2)+1...an−1=\= 0.

目錄 (Table of content)
章節 標題 頁次
致謝(Acknowledgement)…………………..…i
中文摘要(Abstract)……………………….….ii
英文摘要(Abstract)………………………….iii
目錄(Table of content)……………………….iv
第一章 緒論(Introduction)……………………..1
第二章 Toeplitz矩陣(Toeplitz matrices)………3
第三章 特殊矩陣(Special matrices)………….15
參考文獻(References)……………….17
附錄 英文論文

參考文獻(References)
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2.M.T. Chien and F.Z. Lin, On Geometric Properties of the Numerical Range, Linear Algebra and Its Applications, 274(1998),389-410
3.M.T. Chien, On the Numerical Range of Tridiagonal Operators, Linear Algebra and Its Applications, 246(1996),203-214
4.R.A. Horn and C.A. Johnson, Topics in matrix analysis, Cambridge University Press,1991
5.R.A. Horn and C.A. Johnson, Topics in matrix analysis, Cambridge University Press, New York,1985
6.H. Marcus and C. Pesce, Computer generated numerical range and some resulting theorems, Linear and Multilinear Algebra,20(1987),121-157
7.F. Zhang, Matrix Theorem, Sprunger-Verlag, New York,1999