Recent directions in matrix stability

AbstractMatrix stability has been intensively investigated in the past two centuries. We review work that has been done in this topic, focusing on the great progress that has been achieved in the last decade or two. We start with classical stability criteria of Lyapunov, Routh and Hurwitz, and Liénard and Chipart. We then study recently proven sufficient conditions for stability, with particular emphasis on P-matrices. We investigate conditions for the existence of a stable scaling for a given matrix. We review results on other types of matrix stability, such as D-stability, additive D-stability, and Lyapunov diagonal stability. We discuss the weak principal submatrix rank property, shared by Lyapunov diagonally semistable matrices. We also discuss 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