Positive Matrices and Subsidy Allocation Models in Interconnected Industrial Systems

Linear systems with nonnegative or positive coefficients play a central role in the analysis of dynamical processes where admissible states are required to be positive. This paper studies a linear system governed by a positive matrix and interprets it as a spec-tral problem motivated by a subsidy allocation model. The analysis is carried out within the framework of the Perron–Frobenius theory and relies on classical results of linear algebra, in particular Perron’s theorem and Wielandt’s lemma. Using purely theoretical methods, we show that a fair allocation is characterized by a positive eigenvector asso-ciated with the spectral radius of the underlying matrix. The positivity and primitivity of the matrix guarantee the existence and uniqueness of this eigenvector up to scaling, while the convergence of matrix powers ensures the stability of the resulting allocation independently of initial conditions. These results demonstrate that fairness and stability arise as intrinsic consequences of the spectral structure of positive matrices. The paper provides a rigorous mathematical interpretation of equilibrium and stability in linear dynamical systems and illustrates the relevance of positive matrix theory in the study of structured linear models.