کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
9498277 | 1631198 | 2005 | 23 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
A refinement of the split decomposition of a tridiagonal pair
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موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
Let V denote a nonzero finite dimensional vector space over a field K, and let (A, Aâ) denote a tridiagonal pair on V of diameter d. Let V = U0 + â¯Â + Ud denote the split decomposition, and let Ïi denote the dimension of Ui. In this paper, at first we show there exists a unique integer h (0 ⩽ h ⩽ d/2) such that Ïiâ1 < Ïi for 1 ⩽ i ⩽ h, Ïiâ1 = Ïi for h < i ⩽ d â h and Ïiâ1 > Ïi for d â h < i ⩽ d. We call h the height of the tridiagonal pair. For 0 ⩽ r ⩽ h, we define subspaces Ui(r) (r ⩽ i ⩽ d â r) by Ui(r)=Ri-r(Urâ©KerRd-2r+1), where R denotes the rasing map. We show V is decomposed as a direct sum V=âr=0hâi=rd-rUi(r). This gives a refinement of the split decomposition. Define U(r)=âi=rd-rUi(r), and observe V=âr=0hU(r). We show LU(r)âU(r-1)+U(r)+U(r+1) for 0 ⩽ r ⩽ h, where we set U(â1) = U(h+1) = 0. Let F(r):VâU(r) denote the projection. We show the lowering map L is decomposed as L = L(â) + L(0) + L(+), where L(-)=âr=1hF(r-1)LF(r), L(0)=âr=0hF(r)LF(r), and L(+)=âr=0h-1F(r+1)LF(r). These maps satisfy L(-)U(r)âU(r-1),L(0)U(r)âU(r), and L(+)U(r)âU(r+1) for 0 ⩽ r ⩽ h. The main results of this paper are the following: (i) For 0 ⩽ r ⩽ h â 1 and r + 2 ⩽ i ⩽ d â r â 1, RL(+) = αL(+)R holds on Ui(r) for some scalar α; (ii) For 1 ⩽ r ⩽ h and r ⩽ i ⩽ d â r â 1, RL(â) = βL(â)R holds on Ui(r) for some scalar β; (iii) For 0 ⩽ r ⩽ h and r + 1 ⩽ i ⩽ d â r â 1, RL(0) = βL(0)R + γI holds on Ui(r) for some scalars γ, δ. Moreover we give explicit expressions of α, β, γ, δ.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 403, 1 July 2005, Pages 1-23
Journal: Linear Algebra and its Applications - Volume 403, 1 July 2005, Pages 1-23
نویسندگان
Kazumasa Nomura,