Lyapunov's First Theorem

A Necessary and Sufficient condition for all the Eigenvalues of a Real $n\times n$ matrix A to have Negative Real Parts is that the equation

$${\hbox{\sf A}}^{\rm T}{\hbox{\sf V}}+{\hbox{\sf V}}{\hbox{\sf A}}=-{\hbox{\sf I}}$$

has as a solution where ${\hbox{\sf V}}$ is an $n\times n$ matrix and $({\bf x},{\hbox{\sf V}}{\bf x})$ is a positive definite quadratic form.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1122, 1979.