Editing Lyapunov equation (section)

===Continuous time===
Using again the Kronecker product notation and the vectorization operator, one has the matrix equation
:<math> (I_n \otimes A +  \bar{A} \otimes I_n) \operatorname{vec}X = -\operatorname{vec}Q, </math>
where <math>\bar{A}</math> denotes the matrix obtained by complex conjugating the entries of <math>A</math>.

Similar to the discrete-time case, if <math>A</math> is stable (in the sense of [[Hurwitz-stable matrix|Hurwitz stability]], i.e., having eigenvalues with negative real parts), the solution <math>X</math> can also be written as
:<math> X = \int_0^\infty {e}^{A \tau} Q \mathrm{e}^{A^H \tau} d\tau  </math>,
which holds because
:<math> \begin{align}AX+XA^H =& \int_0^\infty A{e}^{A \tau} Q \mathrm{e}^{A^H \tau}+{e}^{A \tau} Q \mathrm{e}^{A^H \tau}A^H d\tau  \\
=&\int_0^\infty \frac{d}{d\tau} {e}^{A \tau} Q \mathrm{e}^{A^H \tau} d\tau \\
=& {e}^{A \tau} Q \mathrm{e}^{A^H \tau} \bigg|_0^\infty\\
=& -Q.
\end{align}
</math>

For comparison, consider the one-dimensional case, where this just says that the solution of <math> 2ax = - q </math> is 
:<math> x = \frac{-q}{2a} = \int_0^{\infty} q{e}^{2 a \tau} d\tau </math>.