The matrix $\displaystyle \mathbf{B}$satifies the following Lyapunov equation

$\displaystyle \begin{gathered}\mathbf{A}^{T}\mathbf{B}\end{gathe red}+\mathbf{BA}=-\mathbf{I}$

prove that necessary and sufficient condition generating a symmetric and positive determined $\displaystyle \mathbf{B}$is that all of the eigen values of $\displaystyle \mathbf{A}$should be negative.

(Hints: rewritten $\displaystyle \mathbf{A}$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!