list manipulation - Eigenvalues and Eigenvectors Order

Say I have a list of eigenvalues and eigenvectors produced from a matrix $M$ using the command {eig1,eig2}=Eigensystem[M], which will return the eigenvalues with respective eigenvectors of $M$.

I want to now produce a new list list1 which will output a list of eigenvalues and the $L^{\infty}$ norms of the respective eigenvectors. How could this be done?

I previously defined Eig1[m_]:=Eigenvalues[m] and Eig2[m_]:=Eigenvectors[m], then used:

EigIf[m_]:=Map[Max[#]&,Eig2[m]]

To then obtain the table with {Eig1[m_],EigIf[m_]}. However, I'm not sure if the order in which the eigenvectors are returned with Eigenvectors[m] is respective to the order given by Eigenvalues[m] (so the first eigenvalue in the latter corresponds to the first eigenvector in Eigenvectors[m]). Can something similar be done with Eigensystem[m]?

Thank you!

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