Toward Solution of Matrix Equation X=Af(X)B+C

TL;DR

This paper investigates the solvability, uniqueness, and solutions of a class of matrix equations involving transpose, conjugate, and Hermitian operations, linking them to standard Stein equations for theoretical and numerical solutions.

Contribution

It establishes the equivalence between the matrix equations' solvability and auxiliary Stein equations, providing closed-form solutions and iterative methods with convergence conditions.

Findings

Equivalence of solvability to standard Stein equations.

Closed-form solutions derived from Stein equation results.

Proposed iterative methods with convergence guarantees.

Abstract

This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation $X=Af(X) B+C$ with $f(X) =X^{\mathrm{T}},$ $f(X) =\bar{X}$ and $f(X) =X^{\mathrm{H}},$ where $X$ is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of $W=\mathcal{A}W\mathcal{B}+\mathcal{C}$ where the dimensions of the coefficient matrices $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ are the same as those of the original equation. Closed-form solutions of equation $X=Af(X) B+C$ can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation equation $X=Af(X) B+C$. Necessary and sufficient conditions are established to guarantee the convergence of the iterations.