Say you have some matrix and the eigenvalues are all the same (degenerate). Let’s say you have $N$ eigenvectors with the label $n$, call them $\psi_n$. If the eigenvalues were not degenerate, then the eigenvectors would form a unique orthonormal basis for some space $\mathbb{V}_N$. However, since they are degenerate, then there is no unique basis because they all have the same eigenvalues, so linear combinations of the eigenvectors are also eigenvectors. My question is:
if we choose any arbitrary basis, will the eigenvectors be orthogonal to each other?
For example, I can choose one eigenvector to be $\psi_n$ and another $\alpha\psi_n + \beta\psi_{n+1}$, and these are already not orthogonal.