Eignproblems Of Two Classes Of Special Matrices Over Max-Plus Algebra

Max-plus algebra operation is a new algorithm produced in modern times.Its main principle is to linearize the nonlinear discrete event dynamic system,so that its state expression has a form similar to the linear system,so as to facilitate the algebraic operation.This paper mainly studies the eigenproblem of adjacency matrix of cartesian product weighted directed graph of two matrices,discusses the algebraic eigenvalue of two matrices and the geometric eigenvalue,and further studies the corresponding main eigenspace.First,in the max-plus algebra operation,the geometric eigenvalue of the adjacency matrix of the Cartesian product-weighted directed graph of two matrices is studied,and the mean value of the maximum circle length is its geometric eigenvalue,also known as the dominant eigenvalue.Using the transformation of the matrix to obtain the principal eigenvalue as the maximum value of the two matrices,we further study the principal eigenspace by taking a vector with zero diagonal terms for each equivalent class in the feature node.The basis of the adjacency matrix of the Cartesian product-weighted directed graph of the two matrices is obtained.Secondly,to study the problem of two max-plus-algebraic matrix tensor product,and then study the corresponding to the main feature space,give the representation of the matrix tensor product of each equivalent class take a diagonal term of zero vector,the vector tensor product,can get the matrix tensor product base representation.At the same time,all the eigenvectors corresponding to their main eigenvalues can be described.Finally,we study algebraic characteristic polynomials of tensor products of max-plus-algebraic matrix.We discuss the weights and roots of its max-plus algebraic characteristic polynomials,aiming to find mutually equivalent max-plus characteristic polynomial functions,while two max-plus polynomials have the same associated polynomial functions if and only if they have the same Newton polygon.The root is calculated using the convexity of max-plus characteristic polynomials,and the root of characteristic polynomials is the algebraic eigenvalue.Finally,it is concluded that even if the positions of two max-plus algebraic matrix tensor product operations are exchanged,their algebraic characteristic polynomials remain equal.