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Suppose matrix $A$ has eigenvalue 1 with corresponding eigenvector $\mathbb{x}$. If $BA$ is to have eigenvalue of 1 with the same eigenvector, what would be the requirement or condition for it?

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    Well, then *also* $\,B\,$ must have the eigenvalue 1 with exactly the same eigenvector $\,x\,$...About your second question: it is a little weird, but in the bottom line is: of course no, as this would contradict directly the definition of eigevector.2012-12-27
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    (For the sake of continuity, the second question was whether a single eigenvector could have multiple eigenvalues)2012-12-27

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