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**ThePerfectHacker** Let $\displaystyle B=\{v_1,v_2,...,v_n\}$ be the set of eigenvectors for $\displaystyle A$ which form a basis for $\displaystyle V$. By definition $\displaystyle Av_j = \ell_j v_j$ for some $\displaystyle \ell_j \in F$. Therefore, $\displaystyle [Av_j]_B = (0,0,...,\ell_j,...0)$ where $\displaystyle \ell_j$ appears in the $\displaystyle j$-coordinate. Therefore, the matrix corresponding to $\displaystyle A$ with respect to this basis is:

$\displaystyle \begin{bmatrix} \ell_1 & 0 & ... & 0 \\ 0 & \ell_2 & ... & 0 \\ ...&...&...&...\\0& 0 & ... & \ell_n \end{bmatrix}$