How Does the Lyapunov Equation Determine Matrix Stability?

[jarvisyang]

The matrix \mathbf{B}satifies the following Lyapunov equation
\begin{gathered}\mathbf{A}^{T}\mathbf{B}\end{gathered}+\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!

 

[ekkilop]

The sufficiency can be obtained by considering
B=\int_{0}^{∞} e^{A^τ t} Q e^{A t} dt

Inserting into the Lyapunov equation gives
AB + BA^{T} = A \int_{0}^{∞} e^{A^τ t} Q e^{A t} dt + \int_{0}^{∞} e^{A^τ t} Q e^{A t} dt A^{T} = \int_{0}^{∞} \frac{d}{dt} (e^{A^τ t} Q e^{A t}) dt = [e^{A^τ t} Q e^{A t}]_{0}^{∞} = -Q
since the eigenvalues of A are negative. Now just let Q=I.