Transpositions matrix

Transpositions matrix (Tr matrix) is square $$n \times n$$ matrix, $$n=2^{m}$$, $$m \in N $$, which elements are obtained from the elements of given n-dimensional vector $$X=(x_i)_{\begin{smallmatrix} i={1,n} \end{smallmatrix}}$$ as follows: $$Tr_{i,j} = x_{(i-1) \oplus (j-1)+1}$$, where $$\oplus$$ 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 $$Tr(X)$$ of order 8, created from arbitrary vector $$X=\begin{pmatrix}x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8 \\\end{pmatrix}$$ $$Tr(X) = \left[\begin{array} {cccc|ccccc} x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & x_8 \\ x_2 & x_1 & x_4 & x_3 & x_6 & x_5 & x_8 & x_7 \\ x_3 & x_4 & x_1 & x_2 & x_7 & x_8 & x_5 & x_6 \\ x_4 & x_3 & x_2 & x_1 & x_8 & x_7 & x_6 & x_5 \\ \hline x_5 & x_6 & x_7 & x_8 & x_1 & x_2 & x_3 & x_4 \\ x_6 & x_5 & x_8 & x_7 & x_2 & x_1 & x_4 & x_3 \\ x_7 & x_8 & x_5 & x_6 & x_3 & x_4 & x_1 & x_2 \\ x_8 & x_7 & x_6 & x_5 & x_4 & x_3 & x_2 & x_1 \end{array}\right] $$

Properties
The figure on the right shows some fours of elements in $$Tr$$ matrix.
 * $$Tr$$ matrix is symmetric matrix.
 * $$Tr$$ matrix is persymmetric matrix, i.e. it is symmetric with respect to the northeast-to-southwest diagonal too.
 * Every one row and column of $$Tr$$ matrix consists all n elements of given vector $$X$$ without repetition.
 * Every two rows $$Tr$$ matrix consists $$n/2$$ fours of elements with the same values of the diagonal elements. In example if $$ Tr_{p,q}$$ and $$ Tr_{u,q}$$ are two arbitrary selected elements from the same column q of $$Tr$$ matrix, then, $$Tr$$ matrix consists one fours of elements $$( Tr_{p,q}, Tr_{u,q}, Tr_{p,v}, Tr_{u,v})$$, for which are satisfied the equations $$ Tr_{p,q}=Tr_{u,v}$$ and $$ Tr_{u,q} = Tr_{p,v}$$. This property, named “Tr-property” is specific to $$Tr$$ matrices.

Transpositions matrix with mutually orthogonal rows (Trs matrix)
The property of fours of $$Tr$$ matrices gives the possibility to create matrix with mutually orthogonal rows and columns ($$Trs$$ matrix ) by changing the sign to an odd number of elements in every one of fours $$( Tr_{p,q}, Tr_{u,q}, Tr_{p,v}, Tr_{u,v})$$, $$p,q,u,v \in [1,n] $$. In [5] is offered algorithm for creating $$Trs$$ matrix using Hadamard product, (denoted by $$ \circ $$) of Tr matrix and n-dimensional Hadamard matrix whose rows (except the first one) are rearranged relative to the rows of Sylvester-Hadamard matrix in order $$R=[1, r_2, \dots, r_n]^T, r_2, \dots, r_n \in [2,n]$$, for which the rows of the resulting Trs matrix are mutually orthogonal.

$$Trs(X) = Tr(X)\circ H(R) $$ $$Trs.{Trs}^T=\parallel X\parallel^2.I_n $$

where:
 * "$$\circ$$" denotes operation Hadamard product
 * $$I_n$$ is n-dimensional Identity matrix.
 * $$H(R)$$ is n-dimensional Hadamard matrix, which rows are interchanged against the Sylvester-Hadamard[4] matrix in given order $$R=[1, r_2, \dots, r_n]^T, r_2, \dots, r_n \in [2,n]$$ for which the rows of the resulting $$Trs$$ matrix are mutually orthogonal.
 * $$X$$ is the vector from which the elements of $$Tr$$ matrix are derived.

Orderings R of Hadamard matrix’s rows were obtained experimentally for $$Trs$$ matrices of sizes 2, 4 and 8. It is important to note, that the ordering R of Hadamard matrix’s rows (against the Sylvester-Hadamard matrix) does not depend on the vector $$X$$. Has been proven[5] that, if $$X$$ is unit vector (i.e. $$\parallel X\parallel=1$$), then $$Trs$$ matrix (obtained as it was described above) is matrix of reflection.

Example of obtaining Trs matrix
Transpositions matrix with mutually orthogonal rows ($$Trs$$ matrix) of order 4 for vector $$X = \begin{pmatrix} x_1, x_2, x_3, x_4 \end{pmatrix}^T$$ is obtained as:

$$Trs(X) = H(R) \circ Tr(X) = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 &-1 & 1 &-1 \\ 1 &-1 &-1 & 1 \\ 1 & 1 &-1 &-1 \\ \end{pmatrix}\circ \begin{pmatrix} x_1 & x_2 & x_3 & x_4 \\ x_2 & x_1 & x_4 & x_3 \\ x_3 & x_4 & x_1 & x_2 \\ x_4 & x_3 & x_2 & x_1 \\ \end{pmatrix}= \begin{pmatrix} x_1 & x_2 & x_3 & x_4 \\ x_2 &-x_1 & x_4 &-x_3 \\ x_3 &-x_4 &-x_1 & x_2 \\ x_4 & x_3 &-x_2 &-x_1 \\ \end{pmatrix} $$ where $$Tr(X)$$ is $$Tr$$ matrix, obtained from vector $$X$$, and "$$\circ$$" denotes operation Hadamard product and $$H(R)$$ is Hadamard matrix, which rows are interchanged in given order $$R$$ for which the rows of the resulting $$Trs$$ matrix are mutually orthogonal. As can be seen from the figure above, the first row of the resulting $$Trs$$ matrix contains the elements of the vector $$X$$ without transpositions and sign change. Taking into consideration that the rows of the $$Trs$$ matrix are mutually orthogonal, we get $$Trs(X).X = \left\| X \right\|^2 \begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix}$$

which means that the $$Trs$$ matrix rotates the vector $$X$$, from which it is derived, in the direction of the coordinate axis $$x_1$$

In [5] are given as examples code of a Matlab functions that creates $$Tr$$ and $$Trs$$ matrices for vector $$X$$ of size n = 2, 4, or, 8. Stay open question is it possible to create $$Trs$$ matrices of size, greater than 8.