Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices

Abstract

In this paper, we use semi-tensor product of quaternion matrices, L-representation of quaternion matrices, and GH-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and L-representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by GH-representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model.

Method for Constructing a Commutative Algebra of Hypercomplex Numbers

“…Algebras of quaternions, octonions, and other hypercomplex numbers enable effective mathematical modeling of various physical processes and distributions of different types of force fields in spaces with different types of symmetry. Therefore, they are widely used to solve various fundamental and applied scientific problems; for example, they are used to solve a number of complex specific problems in the fields of theoretical and applied physics, robotics, cryptography, and digital processing of multidimensional signals [10][11][12][13][14][15][16][17][18][19][20][21][22]. For example, works [10][11][12][13][14][15][16] can be used in applied physics and robotics in the study of complex motions in space; in works [17][18][19][20][21][22], methods for using hypercomplex numbers in solving problems of relativistic physics and encoding multidimensional signals are given.…”