Extension of the discretization bounds to general linear switching systems

Let ${\cal A}$ be a set of $d\times d$ matrices, let $r$ be the maximal spectral radius of the matrices, let $\varepsilon$ be the prescribed accuracy, and let $K(r,\varepsilon)$ denote the critical discretization step size for the associated linear switching system. The preceding bounds were established under the assumption that all matrices in ${\cal A}$ have real spectra. General-spectrum conjecture. Theorem 10 and Corollary 10 remain true for general matrices, without the real-spectra assumption; in particular, the bound

$$K(r,\varepsilon)\geq \frac{6\varepsilon}{\bigl(16d^2-24d+11\bigr)r^2}$$

and the corresponding stability criterion for the discretization step remain valid. This would extend the paper's estimates for the discretization parameter from real-spectrum linear switching systems to arbitrary matrix spectra, a direction identified by the authors as a step toward the general problem.