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 -matrices), the -stability, the additive -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.