| DC 欄位 |
值 |
語言 |
| DC.contributor | 數學系 | zh_TW |
| DC.creator | 呂岳樺 | zh_TW |
| DC.creator | Yue-hua Lu | en_US |
| dc.date.accessioned | 2015-7-7T07:39:07Z | |
| dc.date.available | 2015-7-7T07:39:07Z | |
| dc.date.issued | 2015 | |
| dc.identifier.uri | http://ir.lib.ncu.edu.tw:444/thesis/view_etd.asp?URN=982401004 | |
| dc.contributor.department | 數學系 | zh_TW |
| DC.description | 國立中央大學 | zh_TW |
| DC.description | National Central University | en_US |
| dc.description.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(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. | zh_TW |
| dc.description.abstract | 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. | en_US |
| DC.subject | 數值域 | zh_TW |
| DC.subject | 數值半徑 | zh_TW |
| DC.subject | 張量積 | zh_TW |
| DC.subject | S_n矩陣 | zh_TW |
| DC.subject | 收縮 | zh_TW |
| DC.subject | 友矩陣 | zh_TW |
| DC.subject | numerical range | en_US |
| DC.subject | numerical radius | en_US |
| DC.subject | tensor product | en_US |
| DC.subject | S_n-matrix | en_US |
| DC.subject | contraction | en_US |
| DC.subject | companion matrix | en_US |
| DC.title | Numerical ranges and numerical radii for tensor products of matrices | en_US |
| dc.language.iso | en_US | en_US |
| DC.type | 博碩士論文 | zh_TW |
| DC.type | thesis | en_US |
| DC.publisher | National Central University | en_US |