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**NonCommAlg** if $\displaystyle T$ is diagonalizable and $\displaystyle \lambda_1, \cdots, \lambda_k$ are the distinct eigenvalues of $\displaystyle T,$ then the minimal polynomial of $\displaystyle T$ would be $\displaystyle \prod_{j=1}^k(x-\lambda_j).$ the characteristic polynomial of $\displaystyle T$ in this case would be in the

form $\displaystyle \prod_{j=1}^k (x-\lambda_j)^{n_j},$ where $\displaystyle n_j$ is the number of eigenvalues of $\displaystyle T$ which are equal to $\displaystyle \lambda_j.$

you can find standard facts like this (and much more) about minimal and characteristic polynomial of a linear transformation in any decent linear algebra textbook.