Generalization of the concept of diagonal dominance with applications to matrix D-stability

Some bounds for determinants of relatively $D$-stable matrices

“…. , n. The above inequality shows that A is diagonally C 0 θdominant and, by [12], Theorem 4, relatively D-stable.…”

“…As it was proven in [12] (see [12], Theorem 4), a diagonally C 0 θ -dominant matrix is relatively D-stable. To prove relative additive D-stability, it's enough to notice that if A is diagonally C 0 θ -dominant, then A − D is also diagonally C 0 θ -dominant for any nonnegative diagonal matrix D.…”

Stability and minimal decay rate of systems of ODE

“…Theorem 3 from [37] shows that if a real n × n matrix A is diagonally P (ϵ)-dominant, then it is P (ϵ)-stable, moreover, every matrix of the form DA, where D ∈ D + (0,1] is also P (ϵ)-stable. Basing on Corollary 2, we prove the following D-stability criterion.…”

“…Now let us recall the definition of diagonal dominance with respect to the shifted half-plane C − α , with the shift parameter α ∈ R (see [37]).…”

“…Applying to P (ϵ) the concepts of generalized D-stability and diagonal dominance, developed in [37], we obtain the following definitions. Definition 6. n × n real matrix A is (multiplicative) (P (ϵ), D)-stable if σ(DA) ⊂ P (ϵ) for every positive diagonal matrix D.…”