Determination of a Special Case of Symmetric Matrices and Their Applications

Abstract

This article presents a special case of symmetric matrices, matrices of transpositions (Tr matrices) that are created from the elements of given n-dimensional vector XÎRⁿ, n=2^m, mÎN. Has been proved that Tr matrices are symmetric and persymmetric. Has been proposed algorithm for obtaining matrices of transpositions with mutually orthogonal rows (Trs matrices) of dimensions 2, 4, and 8 as Hadamard product of Tr matrix and matrix of Hadamard and has been investigated their application for QR decomposition and n-dimensional rotation matrix generation. Tests and analysis of the algorithm show that obtaining an orthogonal Trs matrix of sizes 4 and 8 that rotates a given vector to the direction of one of the coordinate axes requires less processing time than obtaining a Housholder matrix of the same size. This makes the Tr and Trs matrices useful in matrix calculations.