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$K$ is an algebraically closed field with characteristic $0$ and $f$ is apolynomial or a holomorphic function. We study all solutions of the equation$XA-AX=f(X)$, in the unknown $X\in M_n(K)$, when $A\in M_n(K)$ isdiagonalizable.

The matrix equation $XA-AX=f(X)$ when $A$ is diagonalizable