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Is it true that for a QM operator in its matrix form must be in the...

Is it true that for a QM operator in its matrix form must be in the same basis that the state its acting on for the matrix multiplication to make sense. For example for me to multiply the matrix form of the Hamiltonian of the system with an eigenfunction of the hamiltonian should all the matrix elements for the hamiltonian operator be made using the eigenbasis of the hamiltonian. If so then please provide a proof. If this isnt the case can you offer a simple example.

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