# A limiting sequence of positive definite matrices

Let $$A\in\mathbb{R}^{n\times n}$$ be a matrix with eigenvalues having (strictly) negative real part. Let $$X\in\mathbb{R}^{n\times n}$$, $$X\succ 0$$, be a positive definite matrix and let $$P\succ 0$$ be the (unique) solution of the following Lyapunov matrix equation $$\tag{1}\label{eq1} AP+PA^\top = - X.$$

My question. Let $$\{X_n\}_{n\ge 0}$$, $$X_n\succ 0$$, be a sequence of positive definite matrices, and let $$\{P_n\}_{n\ge 0}$$, $$P_n\succ 0$$, be the corresponding sequence of solutions of \eqref{eq1}. Suppose that $$\lim_{n\to \infty} X_n=\bar{X}\succeq 0$$ and $$\lim_{n\to \infty} P_n=\bar{P}\succeq 0$$ is singular. I'm wondering whether $$\lim_{n\to \infty} P_n^{-1/2}X_n P_n^{-1/2}$$ always converges to a finite matrix.

A few remarks are in order.

1. If $$\bar{P}$$ is singular, then, $$\bar{X}$$ must be singular as well.
2. If $$A$$ is a scalar matrix, i.e. $$A=\alpha I$$, $$\alpha<0$$, then it is quite easy to see that $$\lim_{n\to \infty} P_n^{-1/2}X_n P_n^{-1/2}$$ converges to a finite matrix.
3. By pre- and post-multiplying \eqref{eq1} by $$P_n^{-1/2}$$, it follows that if $$\lim_{n\to \infty} P_n^{-1/2}A P_n^{1/2}$$ is finite, then $$\lim_{n\to \infty} P_n^{-1/2}X_n P_n^{-1/2}$$ is finite as well.
4. From point 3. it holds $$\mathrm{tr}(P_n^{-1/2}X_n P_n^{-1/2})=-2\,\mathrm{tr}(A)$$ for all $$n$$.