Characteristic Polynomial and Eigenproblem of Triangular Matrix over Interval Min-Plus Algebra

Abstract

A min-plus algebra is a linear algebra over the semiring R_ε', equipped with the operations “"⊕'=min" ” and “⊗=+”. In min-plus algebra, there is the concept of characteristic polynomial obtained from permanent of matrix. Min-plus algebra can be extended to an interval min-plus algebra, which is a set 〖 I(R)〗_ε' equipped with the operations ¯("⊕'" ) and ¯("⊗" ). Matrix over interval min-plus algebra has some special forms, one of which is a triangular matrix. The concept of characteristic polynomial and eigenproblem can be applied to a triangular matrix. There is a more concise formula for solving the eigenproblem of triangular matrix because this matrix is a special form of matrix. This research will discuss the characteristic polynomial and solving eigenproblem of triangular matrix over interval min-plus algebra using its characteristic polynomials. The research method used is a literature study. From the research results, the permanent formula and characteristic polynomial formula of the triangular matrix are obtained. It is also obtained that the smallest corner of the characteristic polynomial is the principal eigenvalue and the vector eigen corresponding to the principal eigenvalue can be obtained through the matrix A_λ. For readers who are interested in this topic, can research about characteristic polynomial and eigenproblem of matrices with other special forms over min-plus interval algebra.

Keywords

• triangular matrix
• smallest corner
• principal eigenvalue
• eigenvector
• characteristic polynomial.