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A generalized eigenvalue problem in the max algebra. (English) Zbl 1121.15011

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.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

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