A modified gradient‐based iterative algorithm for solving the complex conjugate and transpose matrix equations - Long, Yanping; Cui, Jingjing; Huang, Zhengge; Wu, Xiaowen (Mathematical methods in the applied sciences, 2024)
In this paper, we first develop the modified gradient‐based iterative (MGI) method for the complex conjugate and transpose matrix equations A1XB1+A2X‾B2+A3XTB3+A4XHB4=E$$ A_1XB_1+A_2\overline{X}{B}_2+A_3X^TB_3+A_4X^HB_4=E $$. By adopting the updated technique, we can make full use of the latest information to compute the next result, which leads to a faster convergence rate. In theory, we apply the real representation of a complex matrix and the vec‐operator to prove the convergence properties. Furthermore, we extend the MGI algorithm to solve the generalized complex conjugate and transpose matrix equations. Then, the necessary and sufficient conditions for convergence of the MGI algorithm are presented. Lastly, three numerical examples are introduced to testify the efficiency of our methods.