Complete set of invariants
In mathematics, a complete set of invariants for a classification problem is a collection of maps
, such that if and only if for all. In words, such that two objects are equivalent if and only if all invariants are equal.
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 also sufficient for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants, and a complete set of invariants characterizes the coinvariants.
Examples
- In the classification of two-dimensional closed manifolds, Euler characteristic 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 are not.