A generalized eigenvalue problem in the max algebra
Let \(A\) and \(B\) be two (entrywise) nonnegative \({n}\times{n}\) matrices; and define the ``max'' product \(\otimes\) by \((A\otimes{x})_{i}=Max^{n}_{1}a_{im}{x_{m}}\). The generalized eigenvalue problem consists of the problem of the existence and uniqueness of an eigenvalue \(\lambda\) such that \(A\otimes{x}=\lambda{B}\otimes{x}\), \(x\geq{0}\), \({x}\neq{0}\). In this paper, the authors use the results for the case \(A\otimes{x}=\lambda{x}\), \(x\geq{0}\), \({x}\neq{0}\), (when \(B=I\)), to consider the generalized eigenvalue problem: \(A\otimes{x}=\lambda{B}\otimes{x}\), \(x\geq{0}\), \({x}\neq{0}\), where \(A\) and \(B\) are positive \({n}\times{n}\) matrices. The authors explore the possible number of eigenvalues and corresponding eigenvectors when \(A\) and \(B\) are \({2}\times{2}\). They use degree theory and certain geometric properties of the graphs of the some specific functions corresponding to algebraic conditions on certain \({2}\times{2}\) determinants.
Max algebra, generalized eigenvalue problem, Brouwer degree
scientific article; zbMATH DE number 5141619
scientific article