kb/data/en.wikipedia.org/wiki/Complete_set_of_invariants-0.md

title	chunk	source	category	tags	date_saved	instance
Complete set of invariants	1/1	https://en.wikipedia.org/wiki/Complete_set_of_invariants	reference	science, encyclopedia	2026-05-05T07:23:36.291616+00:00	kb-cron

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}.}

== References ==