Finitely Generated Abelian Groups and Similarity of Matrices over a Field / by Christopher Norman
| データ種別 | 電子書籍 |
|---|---|
| 出版者 | London : Springer London |
| 出版年 | 2012 |
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| 書誌ID | OB01024654 |
|---|---|
| 本文言語 | 英語 |
| 一般注記 | Part 1 :Finitely Generated Abelian Groups: Matrices with Integer Entries: The Smith Normal Form -- Basic Theory of Additive Abelian Groups -- Decomposition of Finitely Generated Z-Modules. Part 2: Similarity of Square Matrices over a Field: The Polynomial Ring F[x]and Matrices over F[x]-F[x] Modules: Similarity of t xtMatrices over a Field F -- Canonical Forms and Similarity Classes of Square Matrices over a Field. . ỹ License restrictions may limit access Summary: At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common. However, reduction to Smith normal form, named after its originator H.J.S.Smith in 1861, is a matrix version of the Euclidean algorithm and is exactly what the theory requires in both cases. Starting with matrices over the integers, Part1 of this book provides a measured introduction to such groups: two finitely generated abelian groups are isomorphic if and only if their invariant factor sequences are identical. The analogous theory of matrix similarity over a field is then developed in Part2 starting with matrices having polynomial entries: two matrices over a field are similar if and only if their rational canonical forms are equal. Under certain conditions each matrix is similar to a diagonal or nearly diagonal matrix, namely its Jordan form. The reader is assumed to be familiar with the elementary properties of rings and fields. Also a knowledge of abstract linear algebra including vector spaces, linear mappings, matrices, bases and dimension is essential, although much of the theory is covered in the text but from a more general standpoint: the role of vector spaces is widened to modules over commutative rings. Based on a lecture course taught by the author for nearly thirty years, the book emphasises algorithmic techniques and features numerous worked examples and exercises with solutions. The early chapters form an ideal second course in algebra for second and third year undergraduates. The later chapters, which cover closely related topics, e.g. field extensions, endomorphism rings, automorphism groups, and variants of the canonical forms, will appeal to more advanced students. The book is a bridge between linear and abstract algebra |
| 著者標目 | *Norman, Christopher SpringerLink (Online service) |
| 統一書名標目 | Springer Undergraduate Mathematics Series, |
| 件 名 | LCSH:Mathematics LCSH:Field theory (Physics) LCSH:Group theory LCSH:Matrix theory LCSH:Algorithms FREE:Mathematics FREE:Field Theory and Polynomials FREE:Group Theory and Generalizations FREE:Linear and Multilinear Algebras, Matrix Theory FREE:Algorithms |
| 分 類 | LCC:QA161.A-161.Z LCC:QA161.P59 DC23:512.3 |
| 巻冊次 | ISBN:9781447127307 RefWorks出力（各巻） print ; ISBN:9781447127291 RefWorks出力（各巻） |
| 資料種別 | 機械可読データファイル |
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