Several results on the 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 of the acyclicity of the graph associated with the sign-pattern . Similar results are obtained for the case of  block-matrices and  mixed-matrices, the latter containing both sign patterns and fixed real entries. The problem of assessing the sign-stability of the convex full of a finite  family of Metzler matrices is also solved, and a necessary and sufficient condition for the existence of a common Lyapunov function for all the matrices in the convex hull is obtained. The notion of relative sign-stability is also introduced and a sufficient condition for the relative sign-stability of Metzler matrices is proposed . Several applications of the results are discussed in the last section.
Several results on the 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)-sign-stability, a problem that arises in analysis of certain jump Markov processes. A sufficient condition for the Ker_+(B)-sign-stability of Metzler sign-matrices is obtained and formulated using inverses of sign-matrices and the concept of L^+-matrices . Several applications of the results are discussed in the last section.
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of the
the negativity of the diagonal elements of the Metzler sign-matrix and the
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graph associated with the sign-pattern
associated directed graph
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Metzler
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Metzler
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containing both sign patterns and fixed real
being a class of Metzler matrices containing both sign- and real-type
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full
hull
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and summable
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a common Lyapunov function
common Lyapunov functions
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notion of relative
concept of
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also introduced and a
then generalized to the concept of Ker_+(B)-sign-stability, a problem that arises in analysis of certain jump Markov processes. A
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relative sign-stability of Metzler matrices is proposed
Ker_+(B)-sign-stability of Metzler sign-matrices is obtained and formulated using inverses of sign-matrices and the concept of L^+-matrices