博碩士論文 91221005 詳細資訊




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姓名 李佳萍(Chia-Ping Li)  查詢紙本館藏   畢業系所 數學系
論文名稱 正規壓縮算子與正規延拓算子
(Normal Compressions and Normal Dilations)
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摘要(中) 在此論文中,我們探討「正規壓縮算子」與「正規延拓算子」的性質。在「正規壓縮算子的數值域」(參考文獻8)中有如下的結果:『對於n+1階正規矩陣N的兩個n階正規壓縮算子A與B,A與B么正等價,若且唯若,A與B的所有特徵值都相同(包含重根)』。這篇論文的主要目地則是將上述結果推廣,並分成N是么正矩陣與N是正規矩陣兩種情形來探討。當N是么正矩陣時,A與B么正等價,若且唯若,A與B有超過半數的特徵值相同(包含重根);當N是正規矩陣時,A與B么正等價,若且唯若,A與B有n-1個特徵值相同(包含重根)。
摘要(英) In this thesis, we have two main results. First, we present the n-dimensional compressions of an (n+1)- dimensional unitary matrix are determined, up to unitary equivalence, by only half of their eigenvalues (counting multiplities). Second, we present the n-dimensional compressions of an (n+1)- dimensional normal matrix are determined, up to unitary equivalence, by their n-1 eigenvalues (counting multiplities).
關鍵字(中) ★ 正規延拓算子
★ 正規壓縮算子
關鍵字(英) ★ Normal Compressions
★ Normal Dilations
論文目次 Chapter 1. Introduction …………………………………………………..1
Chapter 2. Notations and Preliminaries ………………………………….3
2.1 Unitary Compressions …………………………………..3
2.2 Normal Compressions …………………………………. 6
Chapter 3. Compression and Dilation ……………………………………11
3.1 Compression……………………………………………. 11
3.2 Dilation ………………………………………………….16
References………………………………………………………………….. 23
參考文獻 (1) M. Adam, J. Maroulas, On compressions of normal matrices, Linear Algebra Appl. 341 (2002) 403--418.
(2) U. Daepp, P. Gorkin, R. Mortini, Ellipses and finite Blaschke products, Amer. Math. Monthly 109 (2002) 785--795.
(3) K. E. Gustafson, D. K. M. Rao, Numerical Range, the Field of Values of Linear Operators and Matrices, Springer, New
York, 1997.
(4) H.-L. Gau, P. Y. Wu, Numerical range of $S(phi)$, Linear and Multilinear Algebra 45 (1998) 49--73.
(5) H.-L. Gau, P. Y. Wu, Dilation to inflations of $S(phi)$, Linear and Multilinear Algebra 45 (1998) 109--123.
(6) H.-L. Gau, P. Y. Wu, Lucas' theorem refined, Linear and Multilinear Algebra 45 (1998) 359--373.
(7) H.-L. Gau, P. Y. Wu, Numerical range and Poncelet property, Taiwanese J. Math. 7 (2003) 173--193.
(8) H.-L. Gau, P. Y. Wu, Numerical range of a normal compression, Linear and Multilinear Algebra. 52 (2004) 195--201.
(9) H.-L. Gau, P. Y. Wu, Numerical range of a normal compression II , Linear and Multilinear Algebra, to appear.
(10) H.-L. Gau, P. Y. Wu, Numerical range circumscribed by two polygons, Linear Algebra Appl. 382 (2004) 155--170.
(11) Halmos, P. R.,A Hilbert space problem book, 2nd end., Springer-Verlag, New York. (1982)
(12) R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.
(13) B. Mirman, V. Borovikov, L. Ladyzhensky, R. Vinograd, Numerical ranges, Poncelet curves, invariant measures,
Linear Algebra Appl. 329 (2001) 61--75.
(14) B. Mirman, P. Y. Wu, Matrix interpretation of Marden's proof of Siebeck's theorem,
preprint.
(15) B. Mirman, Numerical ranges and Poncelet curves, Linear Algebra Appl. 281 (1998) 59--85.
(16) B. Mirman, UB-matrices and conditions for Poncelet polygon to be
closed, Linear Algebra Appl. 360 (2003) 123--150.
(17) P. Y. Wu, Polygons and numerical ranges, Amer. Math. Monthly 107 (2000) 528--540.
指導教授 高華隆(Hwa-Long Gau) 審核日期 2004-6-2
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