Suppose $J $ is a real square matrix and matrix $V = V^\textsf{T}$ is positive-definite. Define $ A = V J $.
Can we show that $J ^\textsf{T} + J$ is negative-definite if, and only if, $A^\textsf{T} P + P A $ is negative-definite for some positive-definite matrix $P = P^\textsf{T}$?
The "if" part is easy. Assume $J ^\textsf{T} + J$ is negative-definite. Let $P = V^{-1}$. Then $A^\textsf{T} P + P A = (VJ)^\textsf{T} V^{-1} + V^{-1} (VJ) = J ^\textsf{T} + J $ is negative-definite.
What about the "only if" part?
Thanks !