An algebraic algorithm for the total least squares problem in commutative quaternionic theory

Abstract

A commutative quaternion total least squares (CQTLS) problem is a method of solving overdetermined sets of linear equations $AX\approx B$ with errors in the matrices A and B. In the theoretical studies and numerical calculations of commutative quaternionic theory, the CQTLS problem is an extremely effective tool for the study of telecommunications, geodesy, and image processing theory. This paper, by means of the real representation of a commutative quaternion matrix, studies the CQTLS problem, derives necessary and sufficient conditions for the CQTLS problem has a commutative quaternion solution, and gives an algebraic algorithm for solving the CQTLS problem.