Transpositions matrix (Tr matrix) is square matrix, , , which elements are obtained from the elements of given n-dimensional vector as follows: , where 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

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The figure below shows Transpositions matrix   of order 8, created from arbitrary vector    

Properties

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  •   matrix is symmetric matrix.
  •   matrix is persymmetric matrix, i.e. it is symmetric with respect to the northeast-to-southwest diagonal too.
  • Every one row and column of   matrix consists all n elements of given vector   without repetition.
  • Every two rows   matrix consists   fours of elements with the same values of the diagonal elements. In example if   and   are two arbitrary selected elements from the same column q of   matrix, then,   matrix consists one fours of elements  , for which are satisfied the equations   and  . This property, named “Tr-property” is specific to   matrices.
 
Fours of elements in Tr matrix

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

Transpositions matrix with mutually orthogonal rows (Trs matrix)

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The property of fours of   matrices gives the possibility to create matrix with mutually orthogonal rows and columns (  matrix ) by changing the sign to an odd number of elements in every one of fours  ,  . In [5] is offered algorithm for creating   matrix using Hadamard product, (denoted by  ) 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  , for which the rows of the resulting Trs matrix are mutually orthogonal.

   

where:

  • " " denotes operation Hadamard product
  •   is n-dimensional Identity matrix.
  •   is n-dimensional Hadamard matrix, which rows are interchanged against the Sylvester-Hadamard[4] matrix in given order   for which the rows of the resulting   matrix are mutually orthogonal.
  •   is the vector from which the elements of   matrix are derived.

Orderings R of Hadamard matrix’s rows were obtained experimentally for   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  . Has been proven[5] that, if   is unit vector (i.e.  ), then   matrix (obtained as it was described above) is matrix of reflection.

Example of obtaining Trs matrix

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Transpositions matrix with mutually orthogonal rows (  matrix) of order 4 for vector   is obtained as:

  where   is   matrix, obtained from vector  , and " " denotes operation Hadamard product and   is Hadamard matrix, which rows are interchanged in given order   for which the rows of the resulting   matrix are mutually orthogonal. As can be seen from the figure above, the first row of the resulting   matrix contains the elements of the vector   without transpositions and sign change. Taking into consideration that the rows of the   matrix are mutually orthogonal, we get  

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

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

See also

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References

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  1. Harville, D. A. (1997). Matrix Algebra from Statistician’s Perspective. Softcover.
  2. Horn, Roger A.; Johnson, Charles R. (2013), Matrix analysis (2nd ed.), Cambridge University Press, ISBN 978-0-521-54823-6
  3. Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486-66434-7
  4. 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. MR 0179093.
  5. 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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