## 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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