### 博碩士論文 92221021 詳細資訊

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(Numerical Ranges of Weighted Permutation Matrices and Weighted Shift Matrices)

 ★ 橢圓形數值域之四階方陣 ★ 數值域邊界上之線段 ★ 正規壓縮算子與正規延拓算子 ★ 可分解友矩陣之數值域 ★ 可分解友矩陣之研究 ★ 關於巴氏空間上連續函數的近乎收斂性 ★ 三角不等式與Jensen不等式之精化 ★ 缺陷指數為1的矩陣之研究 ★ A-Statistical Convergence of Korovkin Type Approximation ★ I-Convergence of Korovkin Type Approximation Theorems for Unbounded Functions ★ 四階方陣的高秩數值域 ★ 位移算子其有限維壓縮算子的反矩陣 ★ 2×2方塊矩陣的數值域 ★ 加權位移矩陣的探討與廣義三角不等式的優化 ★ 喬登方塊和矩陣的張量積之數值域半徑 ★ 3×3矩陣乘積之數值域及數值域半徑

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weighted permutation matrices and weighted shift matrices. Firstly,
we know that if \$A\$ is a \$3 imes3\$ weighted permutation matrix,
\$W(A)\$ has symmetry of \$frac{2pi}{3}\$. Thus, if there is a line
segment on \$partial W(A)\$ then \$W(A)\$ is a triangle. Moreover, \$A\$
is normal. If \$A\$ is a \$3 imes3\$ weighted permutation matrix,
\$W(A)\$ has symmetry of \$frac{2pi}{4}\$. If there is a line segment
on \$partial W(A)\$ then \$W(A)\$ is a quadrangle. Moreover, \$Acong
A_{1}oplus A_{2}\$, where \$A_{1}\$ and \$A_{2}\$ are \$2 imes 2\$
weighted permutation matrices. Let \$A\$ be a \$4 imes4\$ companion
matrix. We will see that \$W(A)\$ has four line segments if and only
if \$A\$ can be reducible.
Another subject is that we are interested in finding the order of
the weights of a weighted shift matrix so that the numerical radius
will be the largest.

★ 加權位移矩陣

★ Weighted Shift Matrix

Abstract (in English) . . . . . . . . . . . . . . . . . . . . . iii
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminaries.. . . . . . . . . . . . . . . . . . . . . .3
2.1Basic Properties of Numerical Range.. . . . . . . . . . . . . . . . . . . . . .3
2.2Weighted Permutation Matrices. . . .. . . . . . . . . . . . . . . . . . . . . . . .4
2.3Companion Matrices .. . . . . . . . . . . . . . . . . . . . . . . .5
2.4Weighted Shift Matrices .. . . . . . . . . . . . . . . . . . . . . . . 6
Chapter 3. Weighted Permutation Matrices and Companion Matrices . . . . . . . . . . . . . . . . . . 10
Chapter 4. Weighted Shift Matrices . . . . . . . . . . . . . . . . . . . . . . 26
References .. . . . . . . . . . . . . . . . . . . . . . 31

nymerical range, Linear Algebra and its Appl., 390 (2004), 75-109.
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and similarity to contractions, Linear Algebra and its Appl., 383 (2004), 127-142.
[3] K. E. Gustafson and D. K. M. Rao, Numerical Range, the Field of Values of
Linear Operators and Matrices, Springer, New York, 1997.
[4] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[5] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University
Press, 1991.
[6] D. S. Keeler, L. Rodman and I. M. Spitkovsky, The Numerical Range of 3 x 3
Matrices, Linear Algebra and its Appl., 252 (1997), 115-139.
[7] A. L. Shields, Weighted Shift Operators and Analytic Function Theory, in Topics in Operator Theory(C. Pearcy, Editor), Math. Surveys, vol. 13, Amer. Math. Soc., Providence, R. I., 1974.
[8] Q. F. Stout, The Numerical Range of a Weighted Shift, Pro. of the Amer. Math. Soc., Vol. 88, No. 3 (1983), 495-502.