Metzlerian and Generalized Metzlerian Matrices: Some Properties and Economic Applications

Abstract

In the first part of the paper we consider the main properties, with respect to stability and existence of solutions of multi-sectoral economic models, of Metzlerian and Morishima matrices. In the second part we introduce various generalized Metzlerian matrices, in order to enlarge the results of Ohyama (1972) in the study of stability and comparative statics for a Walrasian-type equlibrium model.

Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays

“…Generally speaking, Theorem 1 (iv. (3)) is also guaranteed under sufficient conditions if the involved spectral radii [15,25,26] are replaced by matrix norms which are their upperbounds. Also, note that Theorem 2 (ii) has the assumption that ( − ) ∈ × is stable which is needed to guarantee that no eigenvalue crosses the imaginary axis under perturbations of resulting in ( − ) to lose the stability of ( − ) when = 0, i.e., for − = − .…”

“…Theorem A.1 (see [14,15] for any matrix norm ‖ ⋅ ‖, provided that ‖ −1 1 2 ‖ < 1, that is, if ‖ 2 ‖ < = 1/‖ −1 1 ‖ = ( 1 )/‖ 1 ‖, where ( 1 ) = ‖ 1 ‖‖ −1 1 ‖ is the condition number of 1 with respect to the matrix norm ‖ ‖. Property (i) has been proved.…”

“…The noninfective compartment is composed of the subpopulations being free of the disease and typically contains the dynamics of the susceptible and the recovered. For the subsequent study development, the characteristics of the state transition matrices of the linearized infective compartments around the equilibrium points of the epidemic models, which are Metzler matrices, [14,15], as well as their stability properties and the relations of their properties to those of their opposite -matrices, are seen to be crucial mathematical tools. The main reason is that the linearized epidemic system versions around the equilibrium points have also to possess nonnegative solution trajectories since the whole epidemic model has nonnegative solution trajectories under any given nonnegative initial conditions.…”

Decentralized stabilization of fractional positive descriptor continuous-time linear systems

“…Using the known (Boyd et al, 1994;Giorgio and Zuccotti, 2015) procedures we choose the matrices D and N so that…”

“…If (11) holds, then the matrix (10) is an asymptotically stable Metzler matrix. To find the matrices D and N , one of the well-known linear programming or LMI procedures can be used (Boyd et al, 1994;Giorgio and Zuccotti, 2015).…”