3×3矩陣乘積之數值域及數值域半徑

NUMERICAL RANGES AND NUMERICAL RADII OF PRODUCTS OF 3×3 MATRICES

林肯甫(Ken-fu Lin)

指導教授 ： 高華隆

國立中央大學/理學院/數學系/碩士(2015年)

摘要

在本篇論文中，對任意3×3的複數矩陣A和B，我們給出了充分且必要的條件對於AB矩陣乘積的數值域和BA矩陣乘積的數值域相等時。此外，去研究當A和A2的數值域半徑為1且A3的數值域半徑小於1時，A會有什麼樣的矩陣結構。以及最後，我們給出了充分且必要的條件對於當A為壓縮矩陣其特徵值長度皆小於1且A的範數為1，A與B張量積的數值域半徑等於A的範數與B的數值域半徑乘積時。

關鍵字

數值域 ； 數值域半徑 ； 張量積 ； 壓縮矩陣

並列摘要

In this thesis, for any two 3-by-3 complex matrices A and B, we show that the necessary and sufficient conditions for the equality W(AB) = W(BA) to hold, where W() denotes the numerical range of a matrix, and the structure of A when w(A) =w (A2) = 1 and w (A3) < 1, where w() denotes the numerical radius of a matrix, and obtain the necessary and sufficient condition for the equality w(A B) = kAkw(B)to hold when A is a completely nonunitary contraction with kAk = 1, where k k
denotes the usual operator norm of a matrix.

並列關鍵字

numerical range ； numerical radius ； tensor product ； contraction

參考文獻

1. T. Ando, On the structure of operators with numerical radius one, Acta Sci.

3. William F. Donoghue, On the numerical range of a bounded operator, Michigan

4. Miroslav Fiedler, Geometry of the numerical range of matrices, Linear Algebra

products of matrices, Linear and Multilinear Algebra (2013), no. ahead-of-

7. Hwa-Long Gau and Pei Yuan Wu, Finite Blaschke products of contractions,