On The Spectrum For Non-Negative Interval Matrix In Max Algebra

Max algebra is one of the most commonly used tools to research the discrete event systems,such as communication network,traffic control,flexible manufacturing,etc.The time involved in the practical problems is time period in more general cases in our actual life,which is mutative and influenced by some factors outside or inside,rather than a constant value.Therefore,the max algebra using interval as an element provides a effective algebraic method for solving these practical problems.Besides,it is one of the most important research contents to calculate the spectrum of matrix in max algebra.This article mainly study the spectrum of non-negative interval matrix in max algebra from the following several parts.First,we give the definitions and related operations of non-negative interval and nonnegative interval matrix,the partial ordering relation of non-negative are defined,furthermore we give the concepts of the maximum cycle geometric mean interval,spectrum,the digraph and reduced graph about non-negative interval matrix.The next,the relation between maximum cycle geometric mean interval and local spectral radius is discussed through defining the norm-interval of non-negative interval vector and the local spectral radius,thus we obtain an mathematical expression of spectrum with the aid of max cone.Then,the accessible relation in digraph is used and the other mathematical expression of spectrum is obtained.Meanwhile,we can get a simple algorithm about the eigenvalue of arbitrary non-negative matrix.Once again,the Kronecker production of any two non-negative interval matrix are defined,and then we study the relationship between it's eigenvalue and the eigenvalues of the two given matrices,thus we get an expression of the spectrum of Kronecker production.Finally,we define the set of all polynomials with coefficients and variables consisted of non-negative interval in max algebra,and then verify the spectral mapping theorem.