Let $M,N\in \mathbb{C}^{n\times n}$ with $MN = NM$. Can we find polynomials $p,q$ over $\mathbb{C}$ and $C\in \mathbb{C}^{n\times n}$ such that $M = p(C)$ and $N = q(C)$? If $M$ and $N$ are diagonizable we can reduce the problem to diagonal matrices by simultaneous diagonalization. Then the problem is easy. What about the general case? I am also interested in the case where $M$ and $N$ are nilpotent.
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One notable fact: if $M$ is nilpotent of maximal order (i.e. $M^{n-1} \neq 0$), then $MN = NM$ if and only if $N = p(M)$ for some polynomial $p$, which is to say that the condition holds with $C = M$. – Ben Grossmann May 08 '22 at 14:02