Editing Lyapunov equation (section)

{{Short description|Equation from stability analysis}}
The '''Lyapunov equation''', named after the Russian mathematician [[Aleksandr Lyapunov]], is a matrix equation used in the [[Lyapunov stability|stability analysis]] of [[Linear dynamical system|linear dynamical systems]].<ref>{{Cite journal |last=Parks |first=P. C. |date=1992-01-01 |title=A. M. Lyapunov's stability theory—100 years on * |url=https://doi.org/10.1093/imamci/9.4.275 |journal=IMA Journal of Mathematical Control and Information |volume=9 |issue=4 |pages=275–303 |doi=10.1093/imamci/9.4.275 |issn=0265-0754|url-access=subscription }}</ref><ref>{{Cite journal |last=Simoncini |first=V. |author-link=Valeria Simoncini |date=2016-01-01 |title=Computational Methods for Linear Matrix Equations |url=https://epubs.siam.org/doi/10.1137/130912839 |journal=SIAM Review |volume=58 |issue=3 |pages=377–441 |doi=10.1137/130912839 |issn=0036-1445 |hdl-access=free |hdl=11585/586011}}</ref> 

In particular, the '''discrete-time Lyapunov equation''' (also known as '''Stein equation''') for <math>X</math> is 
:<math>A X A^{H} - X + Q = 0</math>
where <math>Q</math> is a [[Hermitian matrix]] and <math>A^H</math> is the [[conjugate transpose]] of <math>A</math>, while the '''continuous-time Lyapunov equation''' is 
:<math>AX + XA^H + Q = 0</math>.