Transpositions matrix

<p>Transpositions matrix (Tr matrix) is square 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>, 
  
    
      
        n
        =
        
          2
          
            m
          
        
      
    
    {\displaystyle n=2^{m}}
  
, 
  
    
      
        m
        ∈
        N
      
    
    {\displaystyle m\in N}
  
, which elements are obtained from the elements of given n-dimensional vector 
  
    
      
        X
        =
        (
        
          x
          
            i
          
        
        
          )
          
            
              
                
                  
                    i
                    =
                    
                      1
                      ,
                      n
                    
                  
                
              
            
          
        
      
    
    {\displaystyle X=(x_{i})_{\begin{smallmatrix}i={1,n}\end{smallmatrix}}}
  
 as follows: 
  
    
      
        T
        
          r
          
            i
            ,
            j
          
        
        =
        
          x
          
            (
            i
            −
            1
            )
            ⊕
            (
            j
            −
            1
            )
            +
            1
          
        
      
    
    {\displaystyle Tr_{i,j}=x_{(i-1)\oplus (j-1)+1}}
  
, where 
  
    
      
        ⊕
      
    
    {\displaystyle \oplus }
  
 denotes operation "<a href="/facts/Bitwise_operation/0gkqnUda">bitwise</a> <a href="/facts/Exclusive_or/FHyr7hGQ">Exclusive or</a>" (XOR). The rows and columns of Transpositions matrix consists <a href="/facts/Permutation/JSw9ukmv">permutation</a> of elements of vector X, as there are <i>n</i>/2 transpositions between every two rows or columns of the matrix
</p>

Transpositions matrix open-in-new

Transpositions matrix