Determination of a Special Case of Symmetric Matrices and Their Applications| Chapter 3 | Current Topics on Mathematics and Computer Science Vol. 6

The elements of a given n-dimensional vector XRn, n=2m, mN are used to build matrices of transpositions (Tr matrices), which are a particular instance of symmetric matrices. Tr matrices have been shown to be both symmetric and persymmetric. As a Hadamard product of Tr matrix and matrix, a new approach for creating transposition matrices with mutually orthogonal rows (Trs matrices) of dimensions 2, 4, and 8 has been presented. Hadamard, and their use in QR decomposition and n-dimensional rotation matrix production has been examined. Obtaining an orthogonal Trs matrix of sizes 4 and 8 that rotates a given vector to the direction of one of the coordinate axes takes less processing time than obtaining a Housholder matrix of the same size, according to tests and analysis of the technique. The Tr and Trs matrices are so useful in matrix calculations.

Author (S) Details

Dr. Ognyan Ivanov Zhelezov
Nikola Vaptsarov Naval Academy, Varna, 9000, Bulgaria.

View Book :- https://stm.bookpi.org/CTMCS-V6/article/view/2583

An Algorithm for Generating N-Dimensional Rotation Matrix | Chapter 2 | Current Topics on Mathematics and Computer Science Vol. 6

This paper describes a new approach for producing an N-dimensional rotation matrix M that rotates a given N-dimensional vector X in the same dimension in the direction of a given N-dimensional vector Y. The Ndimensional Rotation Matrix Generation Algorithm (NRMG) rotates supplied vectors X and Y in the direction of coordinate axis x1 using two-dimensional rotations. Matrix M is generated by multiplying matrix MX by the square root of the square root of the square root of the square root of the square root of the square The inverse of matrix MY, which spins the provided vectors in the axis x1 direction. The Mx and My matrices are not calculated using the RMG algorithm. There is a suggested algorithm for calculating them using rotations in the coordinate planes, but they can alternatively be computed using the Householder transformation, which is more efficient for "dense" vectors. The possibility of executing two-dimensional rotation computations in parallel is also studied.

Author (S) Details

Dr. Ognyan Ivanov Zhelezov
Nikola Vaptsarov Naval Academy, Varna, 9000, Bulgaria.

View Book :- https://stm.bookpi.org/CTMCS-V6/article/view/2582