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!