| 姓名 |
葉鎮宇(Zhen-Yu Ye)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
(Numerical Radii of Matrices and its Submatrices)
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| 相關論文 | |
| 檔案 |
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| 摘要(中) |
假設 A = [a_{ij}]_{i,j=1}^{n} 並且 A′ 是的加權移位矩陣的權重 a_{i,i+1} 對所有 i = 1,...,n
我們知道定理3.1 [1] 的 w(A) ≥ w(A′)。 在這篇論文中，我們考慮何時等式 w(A)= w(A)′ 成立。
在本論文中，我們得到了一些 w(A)= w(A′) 意味著 A = A′ 的矩陣A。 我們證明
(1) 如果 A 是一個非負矩陣，則 w(A)= w(A′) 若且唯若 A = A′，
(2) 如果 A 是一個 Toeplitz 矩陣，則 w(A) = w(A′) 若且唯若
A = A′，以及
(3) 如果 A 是循環矩陣，則 w(A)= w(A′)若且唯若 A = A′。
請注意，A′ 是一個加權移位。 如果 A′ 具有週期性非零權重，我們還考慮何時等式 w(A)= w(A′)
成立。 我們首先研究 A′ 的權重的週期是一。
給出了等式 w(A)= w(A′) 的充分必要條件。
接下來，我們關注 A′ 的權重的週期是偶數。 我們證明，如果
w(A)= w(A′)，那麼A是整體可分解的。 最後，本文還考慮了 A′ 的權重週期為奇數的情況。 |
| 摘要(英) |
Let A=[a_ij]_(i,j=1)^n and A′ be a weighted shift matrix of weights {a_i,i+1} for all i=1,...,n
We know that w(A)≥w(A′) by Proposition 3.1{1}. In this thesis, we consider when the equality w(A)=w(A′) holds. In this thesis, we obtain some classes of matrices A for which w(A)=w(A′) implies A=A′. We show that (1) if A is a non-negative matrix, then w(A)=w(A′) if and only if A=A′, (2) if A is a Toeplitz matrix, then w(A)=w(A′) if and only if A=A′, and (3) if A is a circulant matrix, then w(A)=w(A′) if and only if A=A′.
Note that A′ is a weighted shift. We also consider when the equality w(A)=w(A′) holds if A′ has periodic nonzero weights.
We first study the period of weights of A′ is one.
The sufficient and necessary condition of the equality w(A)=w(A′) is given.
Next, we concerned with the period of weights of A′ is even. We show that if w(A)=w(A′), then A is unitarily reducible. Finally, the case that the period of weights of A′ is odd is also considered in this thesis. |
| 關鍵字(中) |
★ 數值半徑
★ 矩陣 |
關鍵字(英) |
★ Numerical radius
★ Matrices |
| 論文目次 |
Chapter 1. Introduction....................... 1
Chapter 2. Preliminaries...................... 3
2.1 Basic properties of numerical range....... 3
2.2 Basic properties of numerical radius...... 4
2.3 Circulant matrices and Toeplitz matrices.. 5
2.4 Lower bounds for the numerical radius..... 6
2.5 Weighted shift matrices................... 9
Chapter 3. Numerical radius of circulant matrices and Toeplitz matrices.............................. 13
Chapter 4. Weighted shift submatrices with periodic weights........................................ 21
4.1 Period one................................ 21
4.2 Even periodic weights..................... 26
4.3 Odd periodic weights...................... 36
References..................................... 40 |
| 參考文獻 |
[1] H.-L. Gau and P. Y. Wu, Lower bounds for the numerical radius, Oper. Matrices, 11 (2017), 999--1014.
[2] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[3] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
[4] H.-L. Gau, M.-C. Tsai and H.-C. Wang, Weighted shift matrices: unitary equivalence, reducibility and numerical ranges, Linear Algebra Appl., 438 (2013), 498--513.
[5] P. Y. Wu, Numerical Ranges of Hilbert Space Operators, 2013.
[6] K. E. Gustafson and D. K. M. Rao, Numerical Range. The Field of Values of Linear Operators and Matrices, Springer, New York, 1997. |
| 指導教授 |
高華隆(Hwa-Long Gau)
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審核日期 |
2018-6-9 |
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