A lower bound for stability abscissa of a Schur stable matrix
A matrix \(A\) is Schur stable if its spectral radius \(\rho (A)\) is less than one. The stability abscissa of \(A\) is \(\alpha_A = 1 - \rho (A)\). Let \(A\) be Schur stable with real eigenvalues, let \(m(A) = {1 \over n} \text{Tr} (A)\) and let \(s(A) = \sqrt {m (A^2) - m^2 (A)}\). Then \(\rho (A) \leq \sqrt {(1 - \gamma \text{det} ((I - A) (I + A))}\) where \(\gamma = (1 - m (A^2) + {s(A^2) \over \sqrt {n - 1}})^{1 - n} \geq 1\). This gives a lower bound for \(\alpha_A\) which improves a bound established by the first author in an earlier paper.
scientific article; zbMATH DE number 858421