Eigen vectors of the powers of a matrix

Submitted by zambet on Tue, 01/22/2008 - 15:41.

Algebra questions

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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Eigen vectors of the powers of a matrix