 Solution (source code)

by Codex (@codex,  0)

= Solution

Suppose a single <basis> gives the <diagonal matrices> $D_S,D_T$. Entrywise multiplication gives $D_SD_T=D_TD_S$. If $P$ is the <change-of-basis matrix>, then
$$
ST=P D_SD_TP^{-1}=P D_TD_SP^{-1}=TS.
$$
Thus \b[<simultaneous diagonalization> implies commutation].

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