可分解友矩陣之研究

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姓名

孫梅珊(Mei-San Sun) 查詢紙本館藏

畢業系所

數學系

論文名稱

可分解友矩陣之研究
(A Study on Reducible Companion Matrices)

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摘要(中)

本論文探討「可分解友矩陣」的一些性質。我們證明若一個非么正友矩陣 A 可分解成A1 ⊕ A2，則且
。令*1 1 rank( ) 1 kI −AA =* =
2 2 rank( ) 1 n k I AA − − { : rank( * )=1 and | | , ( )} n n Sα ≡ A∈M I−AA λ =α ∀λ ∈σ A
，則相當於1 是屬於A k Sα 且2 是屬於A 1/
n k S α
− 。亦證明每一個屬於n Sα
集合內的矩陣均具有循環、不可分解、且其數值域之邊界為一
可微曲線。並證明下列敘述互為等價：(a) ；(b)
；(c)
1 W(A)=W(A)
1 1) n 2 n1 W(J ) W(A − ⊆ W(A ) W(J ) − ⊆ 。

摘要(英)

In this thesis, we study some properties of reducible companion matrices. We first prove that if a nonunitary reducible companion matrix A is unitarily equivalent to the direct sum A_1oplus A_2 on mathbb{C}^koplusmathbb{C}^{n-k} with sigma(A_1)={aom_n^{j_1},cdots,aom_n^{j_k}} and sigma(A_2)={(1/ ar{a})om_n^{j_{k+1}},cdots,(1/ ar{a})om_n^{j_n}},where |a|>1 and om_n denotes the nth primitive root of 1,then rank(I_k-A^{*}_1A_1)=rank(I_{n-k}-A^{*}_2A_2)=1. We denote mathcal{S}^{al}_nequiv{Ain M_n:rank(I_n-A^{*}A)=1 and |la|=al,forall: lainsigma(A)}, thus A_1 is in mathcal{S}^{al}_k and A_2 is in mathcal{S}^{1/al}_{n-k}. Next, we prove that every mathcal{S}^{al}_n-matrix is irreducible, cyclic, and the boundary of its numerical range is a differentiable curve.
Furthermore, we show that the following statements are equivalent:
(a) W(A)=W(A_1); (b)W(J_{n-1})subseteq W(A_1); (c)W(A_2)subseteq W(J_{n-1}).

關鍵字(中)

★ 可分解之友矩陣

關鍵字(英)

★ reducible companion matrices
★ numerical range

論文目次

1. Introduction . . . . . . . . . . . . . . . . . . . . . . .1
2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Basic Properties of Numerical Range . . . . . . . . . . .3
2.2 Companion Matrices . . . . . . . . 4
3. The S^alpha_n Matrices . . . . . . .6
4. Reducible Companion Matrices . . . . . . .21
References . . . . . . .29

參考文獻

[1] H. Bercovici,(1988). Operator theory and arithmetic in H1, Amer. Math. Soc.,Providence.
[2] H.-L. Gau and P. Y. Wu, Numerical range of S(Á), Linear and Multilinear Al-
gebra, 45 (1998), 49-73.
[3] H.-L. Gau and P. Y.Wu, Lucas' theorem re¯ned, Linear and Multilinear Algebra,45 (1999), 359-373.
[4] H.-L. Gau and P. Y. Wu, Condition for the numerical range to contain an elliptic disc, Linear Algebra and its Appl., 364 (2003), 213-222.
[5] H.-L. Gau and P. Y. Wu, Companion matrices: reducibility, numerical ranges
and similarity to contractions, Linear Algebra and its Appl., 383 (2004), 127-142.
[6] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press,
Cambridge, 1985.
[7] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University
Press, 1991.
[8] D. S. Keeler, L. Rodman and I. M. Spitkovsky, The numerical range of 3 £ 3
matrices, Linear Algebra and its Appl., 252 (1997), 115-139.
[9] B. Mirman, Numerical ranges and Poncelet curves, Linear Algebra and its Appl.,281 (1998), 59-85.
[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

2006-6-29

推文