Suppose we have a diagonalizable matrix $M$ $$M=P D P^{-1}$$ with $D$ diagonal. The usual definition of a function of this matrix, $f(M)$, is $$f(M)=P \text{diag}(f(D_{11}),f(D_{22}),\cdots,f(D_{nn})) P^{-1}$$ Could it be that $f(M)$ is not diagonalizable?
To me, it initially seems that the answer is obviousely no, the diagonalisation of $f(M)$ is right there, it's $P \text{diag}(f(D_{11}),f(D_{22}),\cdots,f(D_{nn})) P^{-1}$. But I remain uncertain. Is this reasoning correct?
A function is just an assignment of a value to every argument. This assignment doesn't need to be in form of an explicit expression, and even when it is, the assignment for a diagonalizable matrix doesn't have to be diagonalizable. For example, let for any diagonalizable matrix $d$ of arbitrary quadratic size, $$f(d) := \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$$ This is not diagonalizable although $d$ is.