Commutation matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> and <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix theory</a>, the commutation matrix is used for transforming the <a href="/facts/Vectorization_(mathematics)/P9a2895l">vectorized</a> form of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> into the vectorized form of its <a href="/facts/Transpose/8wmsagGS">transpose</a>. Specifically, the commutation matrix K(m,n) is the nm × mn <a href="/facts/Permutation_matrix/xC1ran5s">permutation matrix</a> which, for any m × n matrix A, transforms vec(A) into vec(AT):

K(m,n) vec(A) = vec(AT) .
Here vec(A) is the mn × 1 <a href="/facts/Column_vector/pPuRr9Xk">column vector</a> obtain by stacking the columns of A on top of one another:

vec
        ⁡
        (
        
          A
        
        )
        =
        [
        
          
            A
          
          
            1
            ,
            1
          
        
        ,
        …
        ,
        
          
            A
          
          
            m
            ,
            1
          
        
        ,
        
          
            A
          
          
            1
            ,
            2
          
        
        ,
        …
        ,
        
          
            A
          
          
            m
            ,
            2
          
        
        ,
        …
        ,
        
          
            A
          
          
            1
            ,
            n
          
        
        ,
        …
        ,
        
          
            A
          
          
            m
            ,
            n
          
        
        
          ]
          
            
              T
            
          
        
      
    
    {\displaystyle \operatorname {vec} (\mathbf {A} )=[\mathbf {A} _{1,1},\ldots ,\mathbf {A} _{m,1},\mathbf {A} _{1,2},\ldots ,\mathbf {A} _{m,2},\ldots ,\mathbf {A} _{1,n},\ldots ,\mathbf {A} _{m,n}]^{\mathrm {T} }}

where A = [Ai,j]. In other words, vec(A) is the vector obtained by vectorizing A in <a href="/facts/Row-_and_column-major_order/eVYvNWoE">column-major order</a>. Similarly, vec(AT) is the vector obtaining by vectorizing A in row-major order. The <a href="/facts/Cyclic_permutation/UndoYWpl">cycles</a> and other properties of this permutation have been heavily studied for <a href="/facts/In-place_matrix_transposition/suY4JyuC">in-place matrix transposition</a> algorithms.
In the context of <a href="/facts/Quantum_Information/Bl9QIvHj">quantum information theory</a>, the commutation matrix is sometimes referred to as the <a href="/facts/Quantum_logic_gate/EFV5sPCU">swap matrix</a> or swap operator

Commutation matrix open-in-new

Commutation matrix