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# Williamson S.G. Matrix canonical forms

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San Siego: S. Gill Williamson, 2015. — 111 p.
This material is a rewriting of notes handed out by me to beginning graduate students in seminars in combinatorial mathematics (Department of Mathematics, University of California San Diego). Topics covered in this seminar were in algebraic and algorithmic combinatorics. Solid skills in linear and multilinear algebra were required of students in these seminars - especially in algebraic combinatorics. I developed these notes to review the students’ undergraduate linear algebra and improve their proof skills. We focused on a careful development of the general matrix canonical forms as a training ground.
Content:
Functions and Permutations
Algebraic terminology
Sets, lists, multisets and functions
Permutations
Exercises
Matrices and Vector Spaces
Review
Exercises: subspaces
Exercises: spanning sets and dimension
Matrices – basic stuff
Determinants
Elementary properties of determinants
Laplace expansion theorem
Cauchy-Binet theorem
Exercises: Cauchy Binet and Laplace
Hermite/echelon forms
Row equivalence
Hermite form, canonical forms and uniqueness
Stabilizers of GL(n,K); column Hermite forms
Finite dimensional vector spaces and Hermite forms
Diagonal canonical forms – Smith form
Similarity and equivalence
Determinantal divisors and related invariants
Equivalence vs. similarity
Characteristic matrices and polynomials
Rational and Jordan canonical forms.
Index
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