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