Commuting matrices

<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, two <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 are said to commute if 
  
    
      
        A
        B
        =
        B
        A
      
    
    {\displaystyle AB=BA}
  
, or equivalently if their <a href="/facts/Commutator/eYxzFmFY">commutator</a> 
  
    
      
        [
        A
        ,
        B
        ]
        =
        A
        B
        −
        B
        A
      
    
    {\displaystyle [A,B]=AB-BA}
  
 is zero. Matrices 
  
    
      
        A
      
    
    {\displaystyle A}
  
 that commute with matrix 
  
    
      
        B
      
    
    {\displaystyle B}
  
 are called the commutant of matrix 
  
    
      
        B
      
    
    {\displaystyle B}
  
 (and vice versa).
</p><p>A <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of matrices 
  
    
      
        
          A
          
            1
          
        
        ,
        …
        ,
        
          A
          
            k
          
        
      
    
    {\displaystyle A_{1},\ldots ,A_{k}}
  
 is said to commute if they commute pairwise, meaning that every pair of matrices in the set commutes.
</p>

Commuting matrices open-in-new

Commuting matrices