We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation A ⊗ x = λ ⊗ x for usual eigenvectors. Under a suitable assumption, the equation has a nontrivial solution if and only if λ is a root of the characteristic polynomial. (2) The set of algebraic eigenvectors forms a max-plus subspace called algebraic eigenspace. (3) The dimension of each algebraic eigenspace is at most the multiplicity of the corresponding root of the characteristic polynomial.

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关于与最大加特征多项式的根相关的向量

我们讨论最大加代数中的特征值问题。对于最大加方阵，其特征多项式的根不是其特征值。在本文中，我们给出了与特征多项式的根相关联的代数特征向量的概念。代数特征向量在以下三个意义上是常用特征向量的类似物： (1) 代数特征向量满足类似于方程 A ⊗ x = λ ⊗ x 的方程，用于常用特征向量。在适当的假设下，当且仅当 λ 是特征多项式的根时，该方程才有非平凡解。(2) 代数特征向量集形成一个称为代数特征空间的最大加子空间。(3) 每个代数特征空间的维数至多是特征多项式的对应根的重数。