Given the Lyapunov equation below:
$AX + XA^T + B = 0$ with $B=bb^T$
I just want to simulate A $\in \mathcal{M}_{p,p}(\mathbb{R})$ and b $\in \mathbb{R}^{p}$ to find X, solution definite positive.
In theory, having $A$ definite negative and $B$ positive semi definite (by definition : $bb^T$ is positive semi definite) should ensure that $X$ is positive definite.
But it is not the case in practice, so what are the sufficient conditions? Or maybe it is a computational issue: when I check the eigenvalues of $bb^T$, some are negative (but $10^{-11}$ negative).