A General Hermitian Nonnegative-Definite Solution to the Matrix Equation <i>AXB</i> = <i>C</i>

Abstract

We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB C = . Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.

Gradient-Based Iterative Algorithm for a Coupled Complex Conjugate and Transpose Matrix Equations

“…How to solve a matrix equation is a focus question of the applied mathematics and engineering fields [1] [2]. Based on the classical iterative algorithm and the hierarchical identification principle, some gradient-based and least squares based iterative algorithms were established.…”