Eigen vectors of the powers of a matrix

Hello,

Given a matrix-vector pair $ (A, x) $ with $ A \in R^{n \times n} $ and column vector $ x \in R^n $, are there $ k, \lambda $, with $ k \in N^+ $ and $ \lambda \in R $, such that:

$ A^k x = \lambda x $.

Is this problem decidable?

More generally,

For which matrix-vector pairs $ (A, x) $ with $ A \in R^{n \times n} $ and column vector $ x \in R^n $, there exist $ k \in N^+ $ and $ \lambda \in R $ such that:

$ A^k x = \lambda x $.

Thank you,

Maria.

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