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If A and B are conjugate matrices they have the same eigenvectors.

If A and B are conjugate matrices they have the same eigenvectors.

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NO. Conjugate matrices do not have same eigen vector. For that let and are two conjugate matrices. Then there exist an invertible matrix such that   . Now let be an eigen vector of the matrix corresponding to the eigen value . Then we have

This show that is an eigen vector of the matrix corresponding to the same eigen value . Thus conjugate matrices have same eigen value but not necessarily same eigen vector.

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