000 03965nam a22005175i 4500
001 978-1-4471-2730-7
003 DE-He213
005 20140220083236.0
007 cr nn 008mamaa
008 120124s2012 xxk| s |||| 0|eng d
020 _a9781447127307
_9978-1-4471-2730-7
024 7 _a10.1007/978-1-4471-2730-7
_2doi
050 4 _aQA161.A-161.Z
050 4 _aQA161.P59
072 7 _aPBF
_2bicssc
072 7 _aMAT002010
_2bisacsh
082 0 4 _a512.3
_223
100 1 _aNorman, Christopher.
_eauthor.
245 1 0 _aFinitely Generated Abelian Groups and Similarity of Matrices over a Field
_h[electronic resource] /
_cby Christopher Norman.
264 1 _aLondon :
_bSpringer London,
_c2012.
300 _aXII, 381 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringer Undergraduate Mathematics Series,
_x1615-2085
505 0 _aPart 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 xt Matrices over a Field F -- Canonical Forms and Similarity Classes of Square Matrices over a Field.        .
520 _aAt 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.
650 0 _aMathematics.
650 0 _aField theory (Physics).
650 0 _aGroup theory.
650 0 _aMatrix theory.
650 0 _aAlgorithms.
650 1 4 _aMathematics.
650 2 4 _aField Theory and Polynomials.
650 2 4 _aGroup Theory and Generalizations.
650 2 4 _aLinear and Multilinear Algebras, Matrix Theory.
650 2 4 _aAlgorithms.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9781447127291
830 0 _aSpringer Undergraduate Mathematics Series,
_x1615-2085
856 4 0 _uhttp://dx.doi.org/10.1007/978-1-4471-2730-7
912 _aZDB-2-SMA
999 _c100695
_d100695