How Does the Lyapunov Equation Determine Matrix Stability?

[jarvisyang]

The matrix [itex]\mathbf{B}[/itex]satifies the following Lyapunov equation
[tex]\begin{gathered}\mathbf{A}^{T}\mathbf{B}\end{gathered}+\mathbf{BA}=-\mathbf{I}[/tex]
prove that necessary and sufficient condition generating a symmetric and positive determined [itex]\mathbf{B}[/itex]is that all of the eigen values of [itex]\mathbf{A}[/itex]should be negative.
(Hints: rewritten [itex]\mathbf{A}[/itex]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
[itex] B=\int_{0}^{∞} e^{A^τ t} Q e^{A t} dt [/itex]

Inserting into the Lyapunov equation gives
[itex] 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 [/itex]
since the eigenvalues of [itex]A[/itex] are negative. Now just let [itex]Q=I[/itex].