Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation $$ AX+XA^\top=-BB^\top. $$ (N.B. I implicitly assume that such a solution always exists, is unique and positive definite. This holds, for instance, when $A$ is Hurwitz stable, i.e. all its eigenvalues are in the left-half complex plane, and the pair $(A,B)$ is controllable, in control theory jargon.)
Let $\{\lambda_i\}_{i=1}^n$ be the eigenvalues of $X$. After a quick literature overview, I've noticed that several upper bounds on $\lambda_i$'s have been established (see, for instance, [P00], [ASZ02], [BES15] to cite just a few works).
However, what I found missing are lower bounds on $\lambda_i$'s. So, my questions are:
- Are there works in which such lower bounds are discussed?
- If not, why not? Is it due to the fact it is hard to derive "meaningful" lower bounds in case of "general" $A$, $B$?
- If so, it possible to derive lower bounds at least in some special case (e.g., for some particular choice of $A$ and $B$)?
[P00] T. Penzl. "Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case." Systems & Control Letters 40.2 (2000): 139-144.
[ASZ02] A. C. Antoulas, D. C. Sorensen, and Y. Zhou. "On the decay rate of Hankel singular values and related issues." Systems & Control Letters 46.5 (2002): 323-342.
[BES15] J. Baker, M. Embree, and J. Sabino. "Fast singular value decay for Lyapunov solutions with nonnormal coefficients." SIAM Journal on Matrix Analysis and Applications 36.2 (2015): 656-668.
