New proof of the gradient-based iterative algorithm for a complex conjugate and transpose matrix equation

By using the hierarchical identification principle, the gradient-based iterative algorithm is suggested for solving a complex conjugate and transpose matrix equation. With the tools of the real representation of a complex matrix and the vec-operator, a new convergence proof is offered. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the algorithm for any initial iterative matrix. Also a conjecture proposed by Wu et al. (2011) is solved. Two numerical examples are offered to illustrate the effectiveness of the suggested algorithm and conclusions.