Suppose that $A_1,\dots, A_k$ are matrices that any two of them can be approximated by commuting matrices. i.e. for any $i$ and $j$, there are matrices $A_i'$ and $A_j'$ such that $\|A_i-A_i'\|<\varepsilon$ and $\|A_j-A_j'\|<\varepsilon$ and $A_i'A_j'=A_j'A_i'$. Can we find matrices $X_1,\dots,X_k$ such that any two of them commute and $\|X_i-A_i\|<O(\varepsilon)$ for all $i$?

Here the matrix norm could be any unitary invariant norm. I'm specially interested in the operator norm and Hilbert-Schmidt norm.

Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ such that $\|A_i-A_i'\|<\varepsilon$, $\|A_j-A_j'\|<\varepsilon$ and $A_i'A_j'=A_j'A_i'$. Can we find unitary matrices $X_1,\dots,X_k$ such that any two of them commute and $\|X_i-A_i\|<O(\varepsilon)$ for all $i$? What if any three (or small number) of them can be simultaneously approximated by commuting unitary matrices?

Here the matrix norm could be any unitary invariant norm. I'm specially interested in the operator norm and Hilbert-Schmidt norm.

Suppose that $A_1,\dots, A_k$ are matrices that any two of them can be approximated by commuting matrices. i.e. for any $i$ and $j$, there are matrices $A_i'$ and $A_j'$ such that $\|A_i-A_i'\|<\varepsilon$ and $\|A_j-A_j'\|<\varepsilon$ and $A_i'A_j'=A_j'A_i'$. Can we find matrices $X_1,\dots,X_k$ such that any two of them commute and $\|X_i-A_i\|<O(\varepsilon)$ for all $i$?

Here the matrix norm could be any unitary invariant norm. I'm specially interested in the operator norm and Hilert-Schmidt norm.

Suppose that $A_1,\dots, A_k$ are matrices that any two of them can be approximated by commuting matrices. i.e. for any $i$ and $j$, there are matrices $A_i'$ and $A_j'$ such that $\|A_i-A_i'\|<\varepsilon$ and $\|A_j-A_j'\|<\varepsilon$ and $A_i'A_j'=A_j'A_i'$. Can we find matrices $X_1,\dots,X_k$ such that any two of them commute and $\|X_i-A_i\|<O(\varepsilon)$ for all $i$?

Here the matrix norm could be any unitary invariant norm. I'm specially interested in the operator norm and Hilbert-Schmidt norm.