Transpositions matrix (Tr matrix) is square ${\displaystyle n\times n}$ matrix, ${\displaystyle n=2^{m}}$, ${\displaystyle m\in N}$, which elements are obtained from the elements of given n-dimensional vector ${\displaystyle X=(x_{i})_{\begin{smallmatrix}i={1,n}\end{smallmatrix}}}$ as follows: ${\displaystyle Tr_{i,j}=x_{(i-1)\oplus (j-1)+1}}$, 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 ${\displaystyle Tr(X)}$ of order 8, created from arbitrary vector ${\displaystyle X={\begin{pmatrix}x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\\end{pmatrix}}}$

$${\displaystyle Tr(X)=\left[{\begin{array}{cccc|ccccc}x_{1}&x_{2}&x_{3}&x_{4}&x_{5}&x_{6}&x_{7}&x_{8}\\x_{2}&x_{1}&x_{4}&x_{3}&x_{6}&x_{5}&x_{8}&x_{7}\\x_{3}&x_{4}&x_{1}&x_{2}&x_{7}&x_{8}&x_{5}&x_{6}\\x_{4}&x_{3}&x_{2}&x_{1}&x_{8}&x_{7}&x_{6}&x_{5}\\\hline x_{5}&x_{6}&x_{7}&x_{8}&x_{1}&x_{2}&x_{3}&x_{4}\\x_{6}&x_{5}&x_{8}&x_{7}&x_{2}&x_{1}&x_{4}&x_{3}\\x_{7}&x_{8}&x_{5}&x_{6}&x_{3}&x_{4}&x_{1}&x_{2}\\x_{8}&x_{7}&x_{6}&x_{5}&x_{4}&x_{3}&x_{2}&x_{1}\end{array}}\right]}$$

Properties

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

Fours of elements in Tr matrix

The figure on the right shows some fours of elements in ${\displaystyle Tr}$ matrix.

Transpositions matrix with mutually orthogonal rows (Trs matrix)

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

$${\displaystyle Trs(X)=Tr(X)\circ H(R)}$$

$${\displaystyle Trs.{Trs}^{T}=\parallel X\parallel ^{2}.I_{n}}$$

where:

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

Orderings R of Hadamard matrix’s rows were obtained experimentally for ${\displaystyle Trs}$ 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 ${\displaystyle X}$. Has been proven[5] that, if ${\displaystyle X}$ is unit vector (i.e. ${\displaystyle \parallel X\parallel =1}$), then ${\displaystyle Trs}$ matrix (obtained as it was described above) is matrix of reflection.

Example of obtaining Trs matrix

Transpositions matrix with mutually orthogonal rows (${\displaystyle Trs}$ matrix) of order 4 for vector ${\displaystyle X={\begin{pmatrix}x_{1},x_{2},x_{3},x_{4}\end{pmatrix}}^{T}}$ is obtained as:

$${\displaystyle Trs(X)=H(R)\circ Tr(X)={\begin{pmatrix}1&1&1&1\\1&-1&1&-1\\1&-1&-1&1\\1&1&-1&-1\\\end{pmatrix}}\circ {\begin{pmatrix}x_{1}&x_{2}&x_{3}&x_{4}\\x_{2}&x_{1}&x_{4}&x_{3}\\x_{3}&x_{4}&x_{1}&x_{2}\\x_{4}&x_{3}&x_{2}&x_{1}\\\end{pmatrix}}={\begin{pmatrix}x_{1}&x_{2}&x_{3}&x_{4}\\x_{2}&-x_{1}&x_{4}&-x_{3}\\x_{3}&-x_{4}&-x_{1}&x_{2}\\x_{4}&x_{3}&-x_{2}&-x_{1}\\\end{pmatrix}}}$$

where ${\displaystyle Tr(X)}$ is ${\displaystyle Tr}$ matrix, obtained from vector ${\displaystyle X}$, and "${\displaystyle \circ }$" denotes operation Hadamard product and ${\displaystyle H(R)}$ is Hadamard matrix, which rows are interchanged in given order ${\displaystyle R}$ for which the rows of the resulting ${\displaystyle Trs}$ matrix are mutually orthogonal. As can be seen from the figure above, the first row of the resulting ${\displaystyle Trs}$ matrix contains the elements of the vector ${\displaystyle X}$ without transpositions and sign change. Taking into consideration that the rows of the ${\displaystyle Trs}$ matrix are mutually orthogonal, we get

$${\displaystyle Trs(X).X=\left\|X\right\|^{2}{\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}}$$

which means that the ${\displaystyle Trs}$ matrix rotates the vector ${\displaystyle X}$, from which it is derived, in the direction of the coordinate axis ${\displaystyle x_{1}}$

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

See also