Complete set of invariants

In mathematics, a complete set of invariants for a classification problem is a collection of maps

$f_{i}:X\to Y_{i}$

(where $X$ is the collection of objects being classified, up to some equivalence relation $\sim$, and the $Y_{i}$ are some sets), such that $x\sim x'$ if and only if $f_{i}(x)=f_{i}(x')$ for all $i$. 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

$\left(\prod f_{i}\right):(X/\sim )\to \left(\prod Y_{i}\right)$

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

$\prod f_{i}:X\to \prod Y_{i}.$