Solving the system of linear operator equations over generalized bisymmetric matrices

Abstract

This paper proposes an iterative method based on the conjugate gradient method on the normal equations for finding the generalized bisymmetric solution [Formula: see text] to the system of linear operator equations [Formula: see text] where [Formula: see text] are linear operators. By the iterative method, the solvability of this system over the generalized bisymmetric matrix [Formula: see text] can be determined automatically. When the system of linear operator equations is consistent over the generalized bisymmetric matrix [Formula: see text], the iterative method with any generalized bisymmetric initial iterative matrix [Formula: see text] can compute the generalized bisymmetric solution within a finite number of iterations in the absence of roundoff errors. In addition, by the proposed iterative method, the least Frobenius norm generalized bisymmetric solution can be derived when a special initial generalized bisymmetric matrix is chosen. Finally, two numerical examples are presented to support the theoretical results of this paper.