Transpositions matrix
Transpositions matrix (Tr matrix) is square n × n {\displaystyle n\times n} matrix, 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 "bitwise Exclusive or" (XOR). The rows and columns of Transpositions matrix consists permutation of elements of vector X, as there are n/2 transpositions between every two rows or columns of the matrix == Example == The figure below shows Transpositions matrix T r ( X ) {\displaystyle Tr(X)} of order 8, created from arbitrary vector X = ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 ) {\displaystyle X={\begin{pmatrix}x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\\end{pmatrix}}} T r ( X ) = [ x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 2 x 1 x 4 x 3 x 6 x 5 x 8 x 7 x 3 x 4 x 1 x 2 x 7 x 8 x 5 x 6 x 4 x 3 x 2 x 1 x 8 x 7 x 6 x 5 x 5 x 6 x 7 x 8 x 1 x 2 x 3 x 4 x 6 x 5 x 8 x 7 x 2 x 1 x 4 x 3 x 7 x 8 x 5 x 6 x 3 x 4 x 1 x 2 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 ] {\displaystyle Tr(X)=\left[{\begin{array}{cccc|ccccc}x_{1}&x_{2}&x_{3}&x_{4}&x_{5}&x_{6}&x_{7}&x_{8}\\x_{2}&x_{1}&x_{4}&x_{3}&x_{6}&x_{5}&x_{8}&x_{7}\\x_{3}&x_{4}&x_{1}&x_{2}&x_{7}&x_{8}&x_{5}&x_{6}\\x_{4}&x_{3}&x_{2}&x_{1}&x_{8}&x_{7}&x_{6}&x_{5}\\\hline x_{5}&x_{6}&x_{7}&x_{8}&x_{1}&x_{2}&x_{3}&x_{4}\\x_{6}&x_{5}&x_{8}&x_{7}&x_{2}&x_{1}&x_{4}&x_{3}\\x_{7}&x_{8}&x_{5}&x_{6}&x_{3}&x_{4}&x_{1}&x_{2}\\x_{8}&x_{7}&x_{6}&x_{5}&x_{4}&x_{3}&x_{2}&x_{1}\end{array}}\right]} == Properties == T r {\displaystyle Tr} matrix is symmetric matrix.