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Recent directions in matrix stability. (English) Zbl 0759.15010

This is mainly a survey paper on matrix stability. One starts with classical stability criteria. Then one studies the present sufficient conditions for stability (with particular emphasis on P-matrices), the D-stability, the additive D-stability, and the Lyapunov diagonal stability.

One discusses the weak principal submatrix rank property, shared by Lyapunov diagonally semistable matrices, the uniqueness of Lyapunov scaling factors, maximal Lyapunov scaling factors, cones of real positive semidefinite matrices and their applications to matrix stability, and inertia preserving matrices. In this context one derives some original results on stable scaling of complex matrices and on inertia preserving matrices in the acyclic case.

Reviewer: M.Voicu (Iaşi)
MSC:
15A42Inequalities involving eigenvalues and eigenvectors
93E20Optimal stochastic control (systems)
15A48Positive matrices and their generalizations (MSC2000)
15A12Conditioning of matrices
15-02Research monographs (linear algebra)