| 姓名 |
洪炳煌(Bing-Huang Hong)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
可分解友矩陣之數值域
(Numerical Ranges of Reducible Companion Matrices)
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| 相關論文 | |
| 檔案 |
 [Endnote RIS 格式]
[Bibtex 格式]
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| 摘要(中) |
在本論文中，我們首先學習一些關於友矩陣數值域的基本性質。參考文獻1特別探討可分解的友矩陣，同時還證明一個友矩陣的數值域是以原點為圓心的圓盤，其充分必要條件在於這個友矩陣是Jordan區塊。而我們在此僅針對那些數值域為橢圓形的可分解友矩陣作討論，並試圖給這些矩陣一個完整的特徵。
從論文第三節可以看出所有的4 × 4可分解友矩陣將完全被解決，原因是我們會證明一個4 × 4可分解友矩陣的數值域是橢圓，若且為若，這個矩陣的光譜為{a,-a,i/a,-i/a}，其中|a|≧sqrt(1+sqrt(2))；或者這個矩陣的光譜為{a,ai,-1/a,-i/a}，其中|a|≧1+sqrt(2)。最後，我們在論文的第四節就把討論的對象擴大為6 × 6可分解友矩陣。 |
| 摘要(英) |
In this thesis, we study some properties of numerical ranges of companion matrices. Previous works [1] in this respect are the criterion for these matrices to be reducible and show that the numerical range of a companion matrix is a circular disc centered at the origin if and only if the matrix equals the Jordan block. Here we want to give a complete characterization for reducible companion matrices with elliptical numerical range.
In Section 3, 4 × 4 reducible companion matrices will be completely solved. We show that a 4 × 4 reducible companion matrix A has an ellipse as its numerical range if and only if either σ(A)＝{a,-a,i/a,-i/a} where |a|≧sqrt(1+sqrt(2)), or σ(A)＝{a,ai,-1/a,-i/a} where |a|≧1+sqrt(2). Here σ(A) denotes the spectrum of the matrix A. In Section 4, we discuss the cases for 6 × 6 reducible companion matrices. |
| 關鍵字(中) |
★ 可分解的
★ 友矩陣
★ 數值域 |
關鍵字(英) |
★ Companion Matrix
★ Reducible
★ Numerical Range |
| 論文目次 |
1. Introduction..................................................1
2. Preliminaries.................................................3
2.1 Basic Properties of Numerical Ranges.....................3
2.2 Reducible Companion Matrices.............................6
3. 4 × 4 Reducible Companion Matrices............................9
4. 6 × 6 Reducible Companion Matrices...........................16
‧References...................................................20 |
| 參考文獻 |
[1] Hwa–Long Gau and Pei Yuan Wu, Companion matrices: reducibility, numerical ranges and similarity to contractions, Linear Algebra Appl., 383 (2004), 127–142.
[2] U.Haagerup, P. de la Harpe, The numerical radius of a nilpotent operator on a Hilbert space, Proc. Amer. Math. Soc. 115 (1992) 371–379.
[3] R. A. Horn and C. R. Johnson. Matrix analysis, Cambridge University Press, Cambridge, 1985.
[4] R. A. Horn and C. R. Johnson. Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.
[5] D. S. Keeler, L. Rodman and I. M. Spitkovsky, The numerical range of 3 × 3 matrices, Linear Algebra Appl., 252 (1997), 115–139. |
| 指導教授 |
高華隆(Hwa–Long Gau)
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審核日期 |
2006-1-1 |
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