Transpositions Matrix

Transpositions matrix (Tr matrix) is square $n \times n$ matrix, $n=2^$, $m \in N$, which elements are obtained from the elements of given n-dimensional vector $X=(x_i)_$ as follows: $Tr_ = x_$, 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 $X=\beginx_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8 \\\end$
$Tr(X) = \left[\begin x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & x_8 \\ x_2 & x_1 & x_4 & x_3 & x_6 & x_5 & x_8 & x_7 \\ x_3 & x_4 & x_1 & x_2 & x_7 & x_8 & x_5 & x_6 \\ x_4 & x_3 & x_2 & x_1 & x_8 & x_7 & x_6 & x_5 \\ \hline x_5 & x_6 & x_7 & x_8 & x_1 & x_2 & x_3 & x_4 \\ x_6 & x_5 & x_8 & x_7 & x_2 & x_1 & x_4 & x_3 \\ x_7 & x_8 & x_5 & x_6 & x_3 & x_4 & x_1 & x_2 \\ x_8 & x_7 & x_6 & x_5 & x_4 & x_3 & x_2 & x_1 \end\right]$

Properties

* $Tr$ matrix is Symmetric matrix, symmetric matrix.
* $Tr$ matrix is Persymmetric matrix, 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 $Tr_$ and $Tr_$ are two arbitrary selected elements from the same column q of $Tr$ matrix, then, $Tr$ matrix consists one fours of elements $( Tr_, Tr_, Tr_, Tr_)$, for which are satisfied the equations $Tr_=Tr_$ and $Tr_ = Tr_$. This property, named “Tr-property” is specific to $Tr$ matrices.

The figure on the right shows some fours of elements in $Tr$ matrix.

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 $( Tr_, Tr_, Tr_, Tr_)$, p,q,u,v \in ,n. In 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= , r_2, \dots, r_nT , r_2, \dots, r_n \in ,n/math>, for which the rows of the resulting Trs matrix are mutually orthogonal.

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

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 matrix in given order R= , r_2, \dots, r_nT , r_2, \dots, r_n \in ,n/math> 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 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 $X = \begin x_1, x_2, x_3, x_4 \end^T$ is obtained as:

$Trs(X) = H(R) \circ Tr(X) = \begin 1 & 1 & 1 & 1 \\ 1 &-1 & 1 &-1 \\ 1 &-1 &-1 & 1 \\ 1 & 1 &-1 &-1 \\ \end\circ \begin x_1 & x_2 & x_3 & x_4 \\ x_2 & x_1 & x_4 & x_3 \\ x_3 & x_4 & x_1 & x_2 \\ x_4 & x_3 & x_2 & x_1 \\ \end= \begin x_1 & x_2 & x_3 & x_4 \\ x_2 &-x_1 & x_4 &-x_3 \\ x_3 &-x_4 &-x_1 & x_2 \\ x_4 & x_3 &-x_2 &-x_1 \\ \end$
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
$Trs(X).X = \left\, X \right\, ^2 \begin1 \\ 0 \\ 0 \\ 0\end$

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 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.

See also

* Symmetric matrix
* Persymmetric matrix
* Orthogonal matrix

References

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# {{cite book, last=Zhelezov, first=O. I., title=''Determination of a Special Case of Symmetric Matrices and Their Applications'', year=2021, publisher=Current Topics on Mathematics and Computer Science Vol. 6, 29–45, isbn= 978-93-91473-89-1

External links

http://article.sapub.org/10.5923.j.ajcam.20190904.03.html

Matrices