四階方陣的高秩數值域

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姓名

彭煜釗(Yu-Jhau Peng) 查詢紙本館藏

畢業系所

數學系

論文名稱

四階方陣的高秩數值域
(Higher-Rank Numerical Ranges of 4-by-4 Matrices)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 數值域(Numerical Range)
★ 高秩數值域(Higher-Rank Numerical Range)
★ Kippenhahn Curve

關鍵字(英)

★ Kippenhahn Curve
★ Higher-Rank Numerical Range
★ Numerical Range

論文目次

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

參考文獻

[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.

指導教授

高華隆(Hwa-Long Gau)

審核日期

2009-6-17

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