Numerical Radii of Matrices and its Submatrices

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/77813

Title:	Numerical Radii of Matrices and its Submatrices
Authors:	葉鎮宇;Ye, Zhen-Yu
Contributors:	數學系
Keywords:	數值半徑;矩陣;Numerical radius;Matrices
Date:	2018-06-09
Issue Date:	2018-08-31 14:57:48 (UTC+8)
Publisher:	國立中央大學
Abstract:	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。 ;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.
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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