Conditions for unique solvability of the matrix equation AX + X (T) B = C

摘要

The matrix equation AX + XB = C (1) is called the continuous Sylvester equation. Here, A and B are square matrices of possibly different orders m and n, while X and C are mbyn matrices. The famous (continuous) Lyapunov equation AX XA~ T + = C(2) is a particular case of Eq. (1). The equation AXB – X = C (3) is called the discrete Sylvester equation. Here, the matrices are of the same sizes as in (1).