On the $\top$-Stein equation $X=AX^\top B+C$
This work addresses a theoretical gap in matrix equations for mathematicians and engineers working on linear systems, but the results are incremental as they extend known Stein equation theory.
The paper establishes necessary and sufficient conditions for the existence and uniqueness of solutions to the ⊤-Stein equation X = AX^⊤B + C, and provides a numerical algorithm under those conditions.
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.