博碩士論文 972201022 詳細資訊




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姓名 葉雅婷(Ya-ting Yeh)  查詢紙本館藏   畢業系所 數學系
論文名稱 2×2方塊矩陣的數值域
(Numerical Ranges of 2-by-2 Block Matrices)
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摘要(中) 本論文探討對角方塊都是零的2×2方塊矩陣的數值域。我們證明當B是k×k(k>2)矩陣滿足B*B是k-1維的單位矩陣和一維0矩陣的直和,則此2×2方塊矩陣其數值域會是兩個內切在[-1,1]×[-1,1]正方形裡的橢圓的凸包。另一方面,只要B滿足∥B∥=1,我們也對此2×2方塊矩陣其數值域的邊界給出刻劃。此外,對於4階的2×2方塊矩陣 ,我們也給出其數值域會是兩個內切在[-1,1]×[-1,1]正方形裡橢圓的凸包的充分必要條件。
摘要(英) In this thesis, we study the numerical range of a 2-by-2 block matrix with zero diagonal block. We show that if B∈M_(k−1,k) (k ≥ 3) satisfies BB*=I_(k−1), then the numerical range of the 2-by-2 block matrix is the convex hull of two ellipses inscribed in the square [−1, 1] × [−1, 1]. On the other hand, we also show that if B ∈ M_k (k ≥ 3) satisfies
∥B∥=1, then the numerical range of the 2-by-2 block matrix has 4 line segments on its boundary. Among other things, we consider the 2-by-2 block matrix A ∈ M_4, and we give a sufficient and necessary condition in terms of entries of B for numerical range of A being the convex hull of two ellipses.
關鍵字(中) ★ 方塊矩陣
★ 三對角線矩陣
★ 數值域
關鍵字(英) ★ Tridiagonal matrix
★ Numerical range
★ Block matrix
論文目次 Abstract (in English) ii
Contents iii
1 Introduction 1
2 Basic properties for numerical ranges 4
3 Numerical Ranges of 2-by-2 Block Matrices 6
References 34
參考文獻 [1] M.-T. Chien and Hiroshi Nakazato, The numerical range of a tridiagonal
operator, J. Math. Anal. Appl., 373 (2011), 297–304.
[2] H.-L. Gau and P. Y. Wu, Condition for the numerical range to contain an
elliptic disc, Linear Algebra Appl:, 364 (2003), 213–222.
[3] H.-L. Gau and P. Y. Wu, Finite blaschke products of contractions, Linear
Algebra Appl:, 368 (2003), 359–370.
[4] H.-L. Gau and P. Y. Wu, Defect indices of powers of a contraction, Linear
Algebra Appl:, 432 (2010), 2824–2833.
[5] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University
Press, Cambridge, 1991.
[6] D. S. Keeler, L. Rodman and I. M. Spitkovsky, The numerical range of 3×3
matrices, Linear Algebra Appl:, 252(1997), 115–139.
[7] A. Lenard, The numerical range of a pair of projections, Journal of
functional analysis, 10(1972), 410–423
[8] P. Y. Wu, Numerical Ranges of Hilbert Space Operators, preprint.
指導教授 高華隆(Hwa-long Gau) 審核日期 2011-5-31
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