## Prove the Lyapunov equation

The matrix $\mathbf{B}$satifies the following Lyapunov equation
$\begin{gathered}\mathbf{A}^{T}\mathbf{B}\end{gathe red}+\mathbf{BA}=-\mathbf{I}$
prove that necessary and sufficient condition generating a symmetric and positive determined $\mathbf{B}$is that all of the eigen values of $\mathbf{A}$should be negative.
(Hints: rewritten $\mathbf{A}$in the Jordan normal form, one can easily prove the proposition)
But I still cannnot figure it out with the hints!Waiting for your excellent proof!