Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.

Then, they are simultaneously diagonalizable with real entries. BUT, what if I add a small perturbation to them, in the form $A_i +\epsilon B_i$ where $i=1,\cdots, N$, $0<\epsilon \ll1$ and $B_i$'s are arbitrary matrices of $2$-norms being all $1$.

I have an intuition from physics (quantum mechanics) that if $\epsilon$ is sufficiently small compared to $1$ and depending on $N$, the perturbations $\epsilon B_i$ would not hurt the good properties of $A_i$'s. These properties would be simultaneous diagonalizability and functional calculus based on functions of real variables.

However, I cannot find a relevant theorem or reference to this issue.

Could anyone please help me?