On the $\top$-Stein equation $X=AX^\top B+C$

TL;DR

This paper investigates the conditions for existence and uniqueness of solutions to a specific matrix equation involving transpose operations, and proposes a numerical algorithm for solving it.

Contribution

It provides necessary and sufficient conditions for solvability and introduces a new numerical method for solving the $ op$-Stein equation.

Findings

Derived conditions for solution existence and uniqueness.

Developed a numerical algorithm for the $ op$-Stein equation.

Validated the algorithm under solvability conditions.

Abstract

We consider the $\top$-Stein equation $X = AX^\top B + C$, where the operator $(\cdot)^\top$ denotes the transpose ($\top$) of a matrix. In the first part of this paper, we analyze necessary and sufficient conditions for the existence and uniqueness of the solution $X$. In the second part, a numerical algorithm for solving $\top$-Stein equation is given under the solvability conditions.