$A,B$都是n阶方阵，能不能用同一矩阵对角化？

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The minimal polynomial and some applications by Keith Conrad
MSE–Simultaneous diagonalization of commuting linear transformations
The key statement to prove the above theorem is Theorem 4.11 of Keith Conrad's text, which says:

Let $A: V \to V$ be a linear operator. Then $A$ is diagonalizable if and only if its minimal polynomial in $F[T]$ splits in $F[T]$ and has distinct roots.

[$F$ is the ground field, $T$ is an indeterminate, and $V$ is finite dimensional.]
The key point to prove Theorem 4.11 is to check the equality
$$V=E_{\lambda_1}+···+E_{\lambda_r},$$
where the $\lambda_i$ are the distinct eigenvalues and the $E_{\lambda_i}$ are the corresponding eigenspaces. One can prove this by using Lagrange's interpolation formula: put
$$f:=\sum_{i=1}^r\ \prod_{j\not=i}\ \frac{T-\lambda_j}{\lambda_i-\lambda_j}\ \in F[T]$$ and observe that $f(A)$ is the identity of $V$.