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In mathematics, a complete set of invariants for a classification problem is a collection of maps : (where ''X'' is the collection of objects being classified, up to some equivalence relation, and the are some sets), such that if and only if for all ''i''. In words, such that two objects are equivalent if and only if all invariants are equal.〔. See in particular (p. 97 ).〕 Symbolically, a complete set of invariants is a collection of maps such that : is injective. As invariants are, by definition, equal on equivalent objects, equality of invariants is a ''necessary'' condition for equivalence; a ''complete'' set of invariants is a set such that equality of these is ''sufficient'' for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants). ==Examples== * In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) and orientability are a complete set of invariants. * Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「complete set of invariants」の詳細全文を読む スポンサード リンク
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