Lyapunov's Second Theorem

If all the Eigenvalues of a Real Matrix A have Real Parts, then to an arbitrary negative definite quadratic form $({\bf x},{\hbox{\sf W}}{\bf x})$ with ${\bf x}={\bf x}(t)$ there corresponds a positive definite quadratic form $({\bf x},{\hbox{\sf V}}{\bf x})$ such that if one takes

$${d{\bf x}\over dt}={\hbox{\sf A}}{\bf x},$$

then $({\bf x},{\hbox{\sf V}}{\bf x})$ and $({\bf x},{\hbox{\sf W}}{\bf x})$ satisfy

$${d\over dt}({\bf x},{\hbox{\sf V}}{\bf x})=({\bf x},{\hbox{\sf W}}{\bf x}).$$

References

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