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."
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