Stabilizing the Metzler matrices with applications to dynamical systems

• Published in: Calcolo 2020-03, Vol.57 (1), Article 1
• Main Author: Cvetković, Aleksandar
• Format: Article
• Language: English
• Subjects: Algorithms; Dynamical systems; Eigenvalues; Mathematics; Mathematics and Statistics; Matrix methods; Numerical Analysis; Spectral methods; Theory of Computation
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-019-0350-3

Real matrices with non-negative off-diagonal entries play a crucial role for modelling the positive linear dynamical systems. In the literature, these matrices are referred to as Metzler matrices or negated Z-matrices. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in l ∞ , l 1 , and in the max norm. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and numerical experiments for the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered.