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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)
相關論文	★ 橢圓形數值域之四階方陣 ★ 數值域邊界上之線段 ★ 正規壓縮算子與正規延拓算子 ★ 加權排列矩陣及加權位移矩陣之數值域 ★ 可分解友矩陣之數值域 ★ 可分解友矩陣之研究 ★ 關於巴氏空間上連續函數的近乎收斂性 ★ 三角不等式與Jensen不等式之精化 ★ 缺陷指數為1的矩陣之研究 ★ A-Statistical Convergence of Korovkin Type Approximation ★ I-Convergence of Korovkin Type Approximation Theorems for Unbounded Functions ★ 四階方陣的高秩數值域 ★ 位移算子其有限維壓縮算子的反矩陣 ★ 加權位移矩陣的探討與廣義三角不等式的優化 ★ 喬登方塊和矩陣的張量積之數值域半徑 ★ 3×3矩陣乘積之數值域及數值域半徑
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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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