On similarity problem of integral matrices

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姓名	陳穎融(Ying-jung Chen) 查詢紙本館藏	畢業系所	數學系
論文名稱	(On similarity problem of integral matrices)
檔案	[Endnote RIS 格式] [Bibtex 格式] [相關文章] [文章引用] [完整記錄] [館藏目錄] [檢視] [下載] 本電子論文使用權限為同意立即開放。已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。
摘要(中)	在這篇論文中，我們會先介紹矩陣的canonical form，並利用它來解決2-by-2整數矩陣的相似問題。除了canonical forms之外，我們也會找出generators of the stabilizers of the canonical forms，目的是為了解決3-by-3整數矩陣的相似問題。在應用方面，相似問題能幫助我們算出ideal class number of a quadratic algebra over rational number field Q。最後，在論文中也提供了計算canonical form、判斷矩陣是否相似以及算出判斷矩陣是否相似以及算出ideal classes numbers of quadratic Z-orders的程式碼。
摘要(英)	In this thesis, we first introduce the canonical forms and solve the similarity problem of the case of 2-by-2 integral matrices. One direct application is to compute the ideal class number of a quadratic algebra over Q. We also determine the generators of the stabilizers of the canonical forms for 2-by-2 integral matrices, which enables us to solve the similarity problem of 3-by-3 integral matrices with reducible characteristic polynomials. Furthermore, we provide the codes(using Sagemath) for computing the canonical form of a given matrix, determining whether two given matrices are similar or not, and computing the ideal classes numbers of quadratic Z-orders.
關鍵字(中)	★ similarity problem	關鍵字(英)
論文目次	Contents Introduction 1 Chapter I. Preliminaries 5 1. Smith Normal forms over Euclidean domain 5 2. Rational canonical forms 5 3. Block triangular forms over principal ideal domain 6 Chapter II. Similarity of 2-by-2 integral matrices 9 1. Case I 9 2. Case II 11 3. Case III 13 4. Ideal classes of quadratic orders 19 5. Algorithm 22 Chapter III. Similarity of 3-by-3 integral matrices 31 1. Case I 31 2. Case II 33 3. Algorithm 35 Chapter A. Ideal class numbers of quadratic number eld 41 Bibliography 45
參考文獻	[1] Appelgate, H. & Onishi, H., The Similarity Problem for 33 Integer Matrices, Linear algebra and its application 42 (1982) 159-174. [2] Appelgate, H. & Onishi, H., Continued fractions and the conjugacy problem in SL2(Z), Communications in Algebra 9:11, 1121-1130 (1981). [3] Cohen, H., A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer-Verlag Berlin Heidelberg, 1993. [4] Dummit, D. S. & Foote, R. M., Abstract algebra, John Wiley & Sons, 2004. [5] Hardy, G. H. & Wright, E. M., An introduction to the theory of numbers, Oxford University Press, New York, 2008. [6] Ireland, K. & Rosen, M., A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990. [7] Jacobson, N., Basic algebra I, Dover Books in Mathematics, Courier Corporation, 2009. [8] Newman, M., Integral matrices, Pure and applied mathematics, Volume 45, New York and London, 1972. [9] Serre, J. P., A course in arithmetic, Springer, 1973.
指導教授	魏福村陳燕美	審核日期	2018-12-17
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博碩士論文 105221004 詳細資訊