Numerical ranges and numerical radii for tensor products of matrices

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/68861

Title:	Numerical ranges and numerical radii for tensor products of matrices
Authors:	呂岳樺;Lu,Yue-hua
Contributors:	數學系
Keywords:	數值域;數值半徑;張量積;S_n矩陣;收縮;友矩陣;numerical range;numerical radius;tensor product;S_n-matrix;contraction;companion matrix
Date:	2015-07-07
Issue Date:	2015-09-23 14:45:32 (UTC+8)
Publisher:	國立中央大學
Abstract:	在這篇論文中，我們主要討論具備怎樣性質的n×n矩陣A與m×m矩陣B能讓這個等式"w(A\otimes B)=" ‖A‖w(B)成立，其中w(∙)及‖∙‖分別代表一個矩陣的數值半徑(numerical radius)及範數(norm)。我們證明了以下結果：(1)假如A是一個S_n矩陣，則"w(A\otimesB)=" w(B)的充分必要條件是B的數值域(numerical range)是個圓心在原點的圓盤並且k_B≤n，其中k_B這個參數指的是在B的壓縮矩陣中數值域與B相同，這種壓縮矩陣尺寸的最小值；以及(2)若A是個範數為1的completely nonunitary矩陣，而m×m矩陣B滿足k_B=m，則"w(A\otimes B)=" w(B)的充分必要條件是B的數值域是個圓心在原點的圓盤並且k_B≤p_A+1，其中p_A這個參數指的是讓‖A^k ‖=‖A‖^k成立，所有k的最大值。在上述的情況下，我們都得到"A\otimes B" 的數值域與B的數值域相同。接下來，我們也對友矩陣(companion matrix)作一些討論，我們證明：若A是一個n×n的友矩陣，則"W(A\otimes A)" 是個圓心在原點的圓盤的充分必要條件是A是一個n×n的Jordan block J_n.;In this thesis, we characterize matrices A in M_n and B in M_m which yield the equality w(A\otimes B)=\\|A\\|w(B), where w( .) and \\|.\\| denote, respectively, the numerical radius and the operator norm of a matrix. We show that (1) if A is an Sn-matrix, then w(A\otimes B)=w(B) if and only if the numerical range W(B) of B is a circular disc centered at the origin and k_B\leq n, where k_B=min{k:W(V* BV)=W(B) for some V in M_mk with V* V=I_k}; and (2) if A is completely nonunitary with \\|A\\|=1 and k_B =m, then w(A\otimes B)=w(B) if and only if W(B) is a circular disc centered at the origin and k_B\leq pA+1, where p_A=sup{k:\\|A\\|^k=\\|A^k\\|} In the above cases, we all have W(A\otimes B)=W(B). Next, we consider the class of companion matrices. We prove that if A is an n-by-n companion matrix, then W(A\otimes A) is a circular disc centered at the origin if and only if A is equal to the n-by-n Jordan block J_n.
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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