Involutory matrix

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an involutory matrix is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> that is its own <a href="/facts/Inverse_matrix/fPqXk3V8">inverse</a>. That is, multiplication by the matrix 
  
    
      
        
          
            
              A
            
          
          
            n
            ×
            n
          
        
      
    
    {\displaystyle {\mathbf {A}}_{n\times n}}
  
 is an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a> if and only if 
  
    
      
        
          
            
              A
            
          
          
            2
          
        
        =
        
          
            I
          
        
        ,
      
    
    {\displaystyle {\mathbf {A}}^{2}={\mathbf {I}},}
  
 where 
  
    
      
        
          
            I
          
        
      
    
    {\displaystyle {\mathbf {I}}}
  
 is the 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. Involutory matrices are all <a href="/facts/Square_root_of_a_matrix/uWGkQQqN">square roots</a> of the identity matrix. This is a consequence of the fact that any <a href="/facts/Invertible_matrix/fPqXk3V8">invertible matrix</a> multiplied by its inverse is the identity.
</p>

Involutory matrix open-in-new

Involutory matrix