Real iterative algorithms for solving a complex matrix equation with two unknowns

Abstract

This paper discusses two iterative solutions for a complex matrix equation with two unknowns of the form $A_{1}V{B}_1+C_{1}W{D}_1+A_2\overline{V}{B}_2+C_2\overline{W}{D}_2=E$. When this matrix equation is consistent, by using a real inner product in complex matrix space as a tool, solutions for any initial matrices $V_1,W_1$ may be determined in finite iterative steps without round-off errors. In the same equation, we also consider a real iterative algorithm using a real representation of a complex matrix as a tool. Some lemmas and theorems are given and proven in the context of iterative solutions. A numerical example is provided to show the efficacy of the suggested algorithms.

KEYWORDS:

• Complex matrix equation
• iterative algorithm
• inner product
• Frobenius norm
• real representation

Disclosure statement

No potential conflict of interest was reported by the author(s ).