Sign properties of Metzler matrices with applications

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• 作者：Corentin Briat ^{corentin.briat@bsse.ethz.ch corentin@briat.info}
• 关键词：15A48 ; 34D05
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：515
• 期：Complete
• 页码：53-86
• 全文大小：616 K
• 卷排序：515

文摘

Several results about sign properties of Metzler matrices are obtained. It is first established that checking the sign-stability of a Metzler sign-matrix can be either characterized in terms of the Hurwitz stability of the unit sign-matrix in the corresponding qualitative class, or in terms the negativity of the diagonal elements of the Metzler sign-matrix and the acyclicity of the associated directed graph. Similar results are obtained for the case of Metzler block-matrices and Metzler mixed-matrices, the latter being a class of Metzler matrices containing both sign- and real-type entries. The problem of assessing the sign-stability of the convex hull of a finite and summable family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of common Lyapunov functions for all the matrices in the convex hull is obtained. The concept of sign-stability is then generalized to the concept of Ker+(B)Ker+(B)-sign-stability, a problem that arises in the analysis of certain jump Markov processes. A sufficient condition for the Ker+(B)Ker+(B)-sign-stability of Metzler sign-matrices is obtained and formulated using inverses of sign-matrices and the concept of L+L+-matrices. Several applications of the results are discussed in the last section.