Lyapunov equation

The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems.^[1][2]

In particular, the discrete-time Lyapunov equation (also known as Stein equation) for ${\displaystyle X}$ is

${\displaystyle AXA^{H}-X+Q=0}$

where ${\displaystyle Q}$ is a Hermitian matrix and ${\displaystyle A^{H}}$ is the conjugate transpose of ${\displaystyle A}$, while the continuous-time Lyapunov equation is

${\displaystyle AX+XA^{H}+Q=0}$.