Complete set of invariants
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In mathematics, a complete set of invariants for a classification problem is a collection of maps
f i : X → Y i {\displaystyle f_{i}:X\to Y_{i}}
(where X {\displaystyle X} is the collection of objects being classified, up to some equivalence relation ∼ {\displaystyle \sim }, and the Y i {\displaystyle Y_{i}} are some sets), such that x ∼ x ′ {\displaystyle x\sim x'} if and only if f i ( x ) = f i ( x ′ ) {\displaystyle f_{i}(x)=f_{i}(x')} for all i {\displaystyle i}. 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
( ∏ f i ) : ( X / ∼ ) → ( ∏ Y i ) {\displaystyle \left(\prod f_{i}\right):(X/\sim )\to \left(\prod Y_{i}\right)}
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 (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.
- The Jordan normal form of a matrix is a complete invariant for matrices over a field up to conjugation (similarity), but eigenvalues (with multiplicities) are not.
- The elementary divisors are a complete invariant for matrices over a principal ideal domain up to conjugation (or for finitely generated modules over a PID up to isomorphism).
- The signature and rank of a matrix are a complete set of invariants for real symmetric matrices up to congruence (or for real quadratic forms up to equivalence), by Sylvester's law of inertia.
Realizability of invariants
A complete set of invariants does not immediately yield a classification theorem: not all combinations of invariants may be realized. Symbolically, one must also determine the image of
∏ f i : X → ∏ Y i . {\displaystyle \prod f_{i}:X\to \prod Y_{i}.}