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Let $A$ and $B$ $\in M^{n\times n}$ be matrices with $n$ distinct eigenvalues $\{\lambda_1,\ldots ,\lambda_n\}$ and corresponding eigenvectors $\{v_1,v_2,\ldots,v_n\}$. Is it necessary that $A$ and $B$ are equal?

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Yes. Since there are $n$ distinct eigenvalues the corresponding eigenvectors form a basis for $\mathbb R^n$. So knowing the eigenvectors and eigenvalues completely determines what the linear transformation corresponding to the matrix is.

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