跳到主要內容

簡易檢索 / 詳目顯示

回結果列表
研究生： 彭煜釗
Yu-Jhau Peng
論文名稱： 四階方陣的高秩數值域
Higher-Rank Numerical Ranges of 4-by-4 Matrices
指導教授： 高華隆
Hwa-Long Gau
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 97
語文別： 英文
論文頁數： 37
中文關鍵詞： 數值域(Numerical Range) 、高秩數值域(Higher-Rank Numerical Range) 、Kippenhahn Curve
外文關鍵詞： Kippenhahn Curve, Higher-Rank Numerical Range, Numerical Range
相關次數： 點閱：13  下載：0 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 本論文探討一個四階方陣A，其高秩數值域的幾何圖形是什麼樣的圖形。我們將四階方陣的秩二數值域分類。對於一個四階方陣A，我們經由考慮A的associated polynomial來對秩二數值域作分類。對於每一個分類，我們將完整地描述它們的幾何圖形。

    Let $A$ be an $n$-by-$n$ matrix. For $1leq k leq n$, the rank-$k$ numerical range of $A$ is defined and denoted by $Lambda_k(A) = {lambdainmathbb{C}: PAP=lambda P mbox{ for some rank-{it k} orthogonal projection $P$}}$. In this thesis, we give a complete description of the higher-rank numerical ranges of $4$-by-$4$ matrices. We classify the rank-$2$ numerical ranges of $4$-by-$4$ matrices. Our classification is based on the factorability of the associated polynomial $p_A(x,y,z)equiv mathrm{det}(xmathrm{Re,}A + ymathrm{Im,}A + zI_4)$ of a $4$-by-$4$ matrix $A$. For each class, we also completely determine the shape of the rank-$2$ numerical range of a $4$-by-$4$ matrix.

    1 Introduction --1 2 Preliminaries --2 2.1 Basic properties for numerical ranges --2 2.2 Kippenhahn curve --4 2.3 Higher-rank numerical ranges --5 2.4 Numerical ranges and higher-rank numerical ranges of 3 × 3 Matrices --8 3 Higher-Rank Numerical Ranges of 4 × 4 Matrices --9 3.1 Four linear factors --11 3.2 Two linear factors and a quadratic irreducible factor --13 3.3 Two quadratic irreducible factors --16 3.4 A linear factor and a cubic irreducible factor --18 3.5 $p_A$ is irreducible --35 References --37

    [1] M.D. Choi, M. Giesinger, J. A. Holbrook, D.W. Kribs, Geometry of higher-rank numerical ranges, Linear and multilinear Algebra, 56 (2008), 53-64.
    [2] M.D. Choi, J.A. Holbrook, D. W. Kribs, K. Yczkowski, Higher-rank numerical ranges of unitary and normal matrices, Operators and Matrices, 1 (2008), 409-426.
    [3] M.D. Choi, D. W. Kribs, K. Yczkowski, Higher-rank numerical ranges and compression problems, Linear Algebra Appl. 418 (2006), 828-839.
    [4] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, 1985.
    [5] D.S. Keeler, L. Rodman and I.M. Spitkovsky, The numerical range of 3 × 3 matrices, Linear Algebra Appl. 252 (1997), 115-139.
    [6] F. Kirwan, Complex Algebraic Curves, Cambridge Univ. Press, 1992.
    [7] C.K. Li, Y.T. Poon, N.S. Sze, Condition for the higher rank numerical range to be non-empty, Linear and Multilinear Algebra, in press, preprint, http://arxiv.org/abs/0706.1540.
    [8] C.K. Li, N.S. Sze, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, Proc. Amer. Math. Soc. 136 (2008), 3013-3023.
    [9] H.J. Woerdeman, The higher rank numerical range is convex, Linear and Multilinear Algebra, 56 (2008), 65-67.
    [10] P.Y. Wu, Numerical Ranges of Hilbert Space Operators, preprint.

    QR CODE
    :::