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Matrix elements are the components, or entries, of a matrix, typically considered in a certain basis.
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What is the matrix representation of a function of an quantum operator?
It's not clear what you mean by "the function acting on the matrix of the operator."
If you mean
$$f\left(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\right) = \begin{pmatrix}f(a) & f(b) \\ f(c) & f(d)\e …