A note on solutions of the matrix equation AXB=C

TL;DR

This paper analyzes the conditions for the solvability of the matrix equation AXB=C, focusing on the minimal free parameters in its general solution and relating them to matrix ranks using Penrose's and Rohde's formulas.

Contribution

It establishes the relation between free parameters and matrix ranks, and clarifies the minimal parameter count in the general solution of AXB=C.

Findings

Derived necessary and sufficient conditions for solution consistency.

Connected minimal free parameters to ranks of A and B.

Utilized Kronecker product to linearize the matrix equation.

Abstract

This paper deals with necessary and sufficient condition for consistency of the matrix equation $AXB = C$. We will be concerned with the minimal number of free parameters in Penrose's formula $X = A^(1)CB^(1) + Y - A^(1)AYBB^(1)$ for obtaining the general solution of the matrix equation and we will establish the relation between the minimal number of free parameters and the ranks of the matrices A and B. The solution is described in the terms of Rohde's general form of the {1}-inverse of the matrices A and B. We will also use Kronecker product to transform the matrix equation $AXB = C$ into the linear system $(B^T \otimes A)vecX = vec C$.