Characteristic polynomial and eigen problem of diagonal strictly double ℝ-astic matrix over interval min-plus algebra

Max-plus algebra is introduced as an algebraic structure defined over the set ℝε, which is the union of ℝ and {ε}, where ℝ represents the real number system and ε is equal to −∞. It involves two operations: ⊕ and ⊗, where ⊕ denotes the maximum operation and ⊗ represents addition. Min-plus algebra, conversely, represents a different algebraic structure that operates over the set ℝ′ε, which is the union of ℝ and {ε′}, where ℝ represents the real number system and E is equal to −∞. In this case, the ⊕ operation denotes to the minimum operation, while the ⊗ operation still represents addition. The permanent of a matrix is employed to derive the characteristic polynomial in min-plus algebra theory. Solving this polynomial addresses the eigenvalue problem, facilitating the determination of the principal eigenvalue and its corresponding eigenvectors. Derived from min-plus algebra, the concept of interval min-plus algebra is represented in the form of I(ℝ)ε ′ which includes ⊕̄ together with ⊗̄. Within this structure, matrices take on distinct representations, such as the diagonal strictly double ℝ-astic matrix, which enables the application of concepts like characteristic polynomials and eigenvalue problems. This research focuses on the characteristic polynomial and eigenvalue problem for a diagonal strictly double ℝ-astic matrix within the context of interval min-plus algebra. The methodology for the study is based on a literature review, which results in the characteristic polynomial formula where the smallest corner is equal to the principal eigenvalue, and the corresponding eigenvector is represented by the columns of Aλ, with its diagonal elements being [0,0].