Numerical ranges and numerical radii for tensor products of matrices

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姓名

呂岳樺(Yue-hua Lu) 查詢紙本館藏

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數學系

論文名稱

(Numerical ranges and numerical radii for tensor products of matrices)

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摘要(中)

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(Aotimes 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(AotimesB)=" 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(Aotimes 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"Aotimes 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(Aotimes 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.

摘要(英)

In this thesis, we characterize matrices A in M_n and B in M_m which yield the equality w(Aotimes 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(Aotimes B)=w(B) if and only if the numerical range W(B) of B is a circular disc
centered at the origin and k_Bleq 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(Aotimes B)=w(B) if and only if W(B) is a circular disc centered at the origin and k_Bleq pA+1,
where p_A=sup{k:|A|^k=|A^k|}
In the above cases, we all have W(Aotimes 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(Aotimes 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.

關鍵字(中)

★ 數值域
★ 數值半徑
★ 張量積
★ S_n矩陣
★ 收縮
★ 友矩陣

關鍵字(英)

★ numerical range
★ numerical radius
★ tensor product
★ S_n-matrix
★ contraction
★ companion matrix

論文目次

1 Introduction and Preliminaries..........................1
1.1 Numerical ranges and numerical radii..................1
1.2 Sn-matrices...........................................4
1.3 Tensor product........................................7
2 Numerical radii and numerical ranges for tensor products of matrices..............................................13
2.1 Sn-matrices..........................................13
2.2 Contractions.........................................21
3 Companion Matrices.....................................33
3.1 Introduction.........................................33
3.2 Tensor Products of Companion Matrices................36
Bibliography.............................................51

參考文獻

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指導教授

高華隆(Hwa-Long Gau)

審核日期

2015-7-7

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