Finitely Generated Abelian Groups and Similarity of Matrices over a Field

Nonfiction, Science & Nature, Mathematics, Algebra, Computers, Programming
Cover of the book Finitely Generated Abelian Groups and Similarity of Matrices over a Field by Christopher Norman, Springer London
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Christopher Norman ISBN: 9781447127307
Publisher: Springer London Publication: January 25, 2012
Imprint: Springer Language: English
Author: Christopher Norman
ISBN: 9781447127307
Publisher: Springer London
Publication: January 25, 2012
Imprint: Springer
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Springer London

Cover of the book Multimodality Imaging for Transcatheter Aortic Valve Replacement by Christopher Norman
Cover of the book Inflammatory Response in Cardiovascular Surgery by Christopher Norman
Cover of the book Fundamentals of Digital Imaging in Medicine by Christopher Norman
Cover of the book A History of Endometriosis by Christopher Norman
Cover of the book Guide to Computer Network Security by Christopher Norman
Cover of the book Understanding Mechanical Ventilation by Christopher Norman
Cover of the book How to Observe the Sun Safely by Christopher Norman
Cover of the book Exercise Cardiopulmonary Function in Cardiac Patients by Christopher Norman
Cover of the book Supporting People with Dementia Using Pervasive Health Technologies by Christopher Norman
Cover of the book Paul Lévy and Maurice Fréchet by Christopher Norman
Cover of the book S-Variable Approach to LMI-Based Robust Control by Christopher Norman
Cover of the book Controversies and Innovations in Urological Surgery by Christopher Norman
Cover of the book Sustainable Design by Christopher Norman
Cover of the book Hypermobility of Joints by Christopher Norman
Cover of the book Systems Approaches to Managing Change: A Practical Guide by Christopher Norman
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy