Springer Undergraduate Mathematics Series

Finitely Generated Abelian Groups and Similarity of Matrices over a Field

Authors: Norman, Christopher

  • The theory of finitely generated abelian groups is introduced in an understandable and concrete way
  • The analogous theory of similarity of square matrices over a field, including the Jordan form, is explained step by step
  • Manipulative techniques are stressed to get students ‘on board’
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About this Textbook

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, Part 1 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 Part 2 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.

Reviews

From the reviews:

“‘Designed to be a second course in linear algebra suitable for second/third mathematics undergraduates, or postgraduates’, will help the readers to improve their knowledge of basic notions in algebra, as the structure of finitely generated Abelian groups or canonical forms of square matrices. The book contains detailed proofs, illuminating examples, and many exercises which will help the reader to understand very well the presented techniques. … will help the reader to pass from linear to abstract algebra, so I recommend it to all undergraduate students.” (Simion Sorin Breaz, Zentralblatt MATH, Vol. 1242, 2012)


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Table of contents (6 chapters)

  • Matrices with Integer Entries: The Smith Normal Form

    Norman, Christopher

    Pages 9-46

  • Basic Theory of Additive Abelian Groups

    Norman, Christopher

    Pages 47-96

  • Decomposition of Finitely Generated ℤ-Modules

    Norman, Christopher

    Pages 97-152

  • The Polynomial Ring F[x] and Matrices over F[x]

    Norman, Christopher

    Pages 165-202

  • F[x]-Modules: Similarity of t×t Matrices over a Field F

    Norman, Christopher

    Pages 203-249

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-1-4471-2730-7
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.95
price for USA
  • ISBN 978-1-4471-2729-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Finitely Generated Abelian Groups and Similarity of Matrices over a Field
Authors
Series Title
Springer Undergraduate Mathematics Series
Copyright
2012
Publisher
Springer-Verlag London
Copyright Holder
Springer-Verlag London Limited
eBook ISBN
978-1-4471-2730-7
DOI
10.1007/978-1-4471-2730-7
Softcover ISBN
978-1-4471-2729-1
Series ISSN
1615-2085
Edition Number
1
Number of Pages
XII, 381
Topics