Transpositions matrix
Transpositions matrix (Tr matrix) is square [math]\displaystyle{ n \times n }[/math] matrix, [math]\displaystyle{ n=2^{m} }[/math], [math]\displaystyle{ m \in N }[/math], which elements are obtained from the elements of given n-dimensional vector [math]\displaystyle{ X=(x_i)_{\begin{smallmatrix} i={1,n} \end{smallmatrix}} }[/math] as follows: [math]\displaystyle{ Tr_{i,j} = x_{(i-1) \oplus (j-1)+1} }[/math], where [math]\displaystyle{ \oplus }[/math] 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 [math]\displaystyle{ Tr(X) }[/math] of order 8, created from arbitrary vector [math]\displaystyle{ X=\begin{pmatrix}x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8 \\\end{pmatrix} }[/math] [math]\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] }[/math]
Properties
- [math]\displaystyle{ Tr }[/math] matrix is symmetric matrix.
- [math]\displaystyle{ Tr }[/math] matrix is persymmetric matrix, i.e. it is symmetric with respect to the northeast-to-southwest diagonal too.
- Every one row and column of [math]\displaystyle{ Tr }[/math] matrix consists all n elements of given vector [math]\displaystyle{ X }[/math] without repetition.
- Every two rows [math]\displaystyle{ Tr }[/math] matrix consists [math]\displaystyle{ n/2 }[/math] fours of elements with the same values of the diagonal elements. In example if [math]\displaystyle{ Tr_{p,q} }[/math] and [math]\displaystyle{ Tr_{u,q} }[/math] are two arbitrary selected elements from the same column q of [math]\displaystyle{ Tr }[/math] matrix, then, [math]\displaystyle{ Tr }[/math] matrix consists one fours of elements [math]\displaystyle{ ( Tr_{p,q}, Tr_{u,q}, Tr_{p,v}, Tr_{u,v}) }[/math], for which are satisfied the equations [math]\displaystyle{ Tr_{p,q}=Tr_{u,v} }[/math] and [math]\displaystyle{ Tr_{u,q} = Tr_{p,v} }[/math]. This property, named “Tr-property” is specific to [math]\displaystyle{ Tr }[/math] matrices.
The figure on the right shows some fours of elements in [math]\displaystyle{ Tr }[/math] matrix.
Transpositions matrix with mutually orthogonal rows (Trs matrix)
The property of fours of [math]\displaystyle{ Tr }[/math] matrices gives the possibility to create matrix with mutually orthogonal rows and columns ([math]\displaystyle{ Trs }[/math] matrix ) by changing the sign to an odd number of elements in every one of fours [math]\displaystyle{ ( Tr_{p,q}, Tr_{u,q}, Tr_{p,v}, Tr_{u,v}) }[/math], [math]\displaystyle{ p,q,u,v \in [1,n] }[/math]. In [5] is offered algorithm for creating [math]\displaystyle{ Trs }[/math] matrix using Hadamard product, (denoted by [math]\displaystyle{ \circ }[/math]) 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 [math]\displaystyle{ R=[1, r_2, \dots, r_n]^T , r_2, \dots, r_n \in [2,n] }[/math], for which the rows of the resulting Trs matrix are mutually orthogonal.
[math]\displaystyle{ Trs(X) = Tr(X)\circ H(R) }[/math] [math]\displaystyle{ Trs.{Trs}^T=\parallel X\parallel^2.I_n }[/math]
where:
- "[math]\displaystyle{ \circ }[/math]" denotes operation Hadamard product
- [math]\displaystyle{ I_n }[/math] is n-dimensional Identity matrix.
- [math]\displaystyle{ H(R) }[/math] is n-dimensional Hadamard matrix, which rows are interchanged against the Sylvester-Hadamard[4] matrix in given order [math]\displaystyle{ R=[1, r_2, \dots, r_n]^T , r_2, \dots, r_n \in [2,n] }[/math] for which the rows of the resulting [math]\displaystyle{ Trs }[/math] matrix are mutually orthogonal.
- [math]\displaystyle{ X }[/math] is the vector from which the elements of [math]\displaystyle{ Tr }[/math] matrix are derived.
Orderings R of Hadamard matrix’s rows were obtained experimentally for [math]\displaystyle{ Trs }[/math] 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 [math]\displaystyle{ X }[/math]. Has been proven[5] that, if [math]\displaystyle{ X }[/math] is unit vector (i.e. [math]\displaystyle{ \parallel X\parallel=1 }[/math]), then [math]\displaystyle{ Trs }[/math] matrix (obtained as it was described above) is matrix of reflection.
Example of obtaining Trs matrix
Transpositions matrix with mutually orthogonal rows ([math]\displaystyle{ Trs }[/math] matrix) of order 4 for vector [math]\displaystyle{ X = \begin{pmatrix} x_1, x_2, x_3, x_4 \end{pmatrix}^T }[/math] is obtained as:
[math]\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} }[/math] where [math]\displaystyle{ Tr(X) }[/math] is [math]\displaystyle{ Tr }[/math] matrix, obtained from vector [math]\displaystyle{ X }[/math], and "[math]\displaystyle{ \circ }[/math]" denotes operation Hadamard product and [math]\displaystyle{ H(R) }[/math] is Hadamard matrix, which rows are interchanged in given order [math]\displaystyle{ R }[/math] for which the rows of the resulting [math]\displaystyle{ Trs }[/math] matrix are mutually orthogonal. As can be seen from the figure above, the first row of the resulting [math]\displaystyle{ Trs }[/math] matrix contains the elements of the vector [math]\displaystyle{ X }[/math] without transpositions and sign change. Taking into consideration that the rows of the [math]\displaystyle{ Trs }[/math] matrix are mutually orthogonal, we get [math]\displaystyle{ Trs(X).X = \left\| X \right\|^2 \begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix} }[/math]
which means that the [math]\displaystyle{ Trs }[/math] matrix rotates the vector [math]\displaystyle{ X }[/math], from which it is derived, in the direction of the coordinate axis [math]\displaystyle{ x_1 }[/math]
In [5] are given as examples code of a Matlab functions that creates [math]\displaystyle{ Tr }[/math] and [math]\displaystyle{ Trs }[/math] matrices for vector [math]\displaystyle{ X }[/math] of size n = 2, 4, or, 8. Stay open question is it possible to create [math]\displaystyle{ Trs }[/math] matrices of size, greater than 8.
See also
References
- Harville, D. A. (1997). Matrix Algebra from Statistician’s Perspective. Softcover.
- Horn, Roger A.; Johnson, Charles R. (2013), Matrix analysis (2nd ed.), Cambridge University Press, ISBN 978-0-521-54823-6
- Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486-66434-7, https://books.google.com/books?id=ULMmheb26ZcC&dq=linear+algebra+determinant&pg=PA1
- Baumert, L. D.; Hall, Marshall (1965). "Hadamard matrices of the Williamson type". Math. Comp. 19 (91): 442–447. doi:10.1090/S0025-5718-1965-0179093-2.
- Zhelezov, O. I. (2021). Determination of a Special Case of Symmetric Matrices and Their Applications. Current Topics on Mathematics and Computer Science Vol. 6, 29–45. ISBN 978-93-91473-89-1.
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