I am looking for a meaningful way to distinguish, from the point of view of notation, between eigenvalues counted with and without multiplicity.
Given a $d$-dimensional vector space $V$ and a diagonalizable linear map $V \to V$, I am denoting the eigenvalues by $\lambda_{1}, \dotsc, \lambda_{k}$, where $k \leq d$ because an eigenvalue may have multiplicity greater than one.
On the other hand, I also need to refer to a basis of eigenvectors $e_{1}, \dotsc, e_{d}$; for each $e_{j}$, I need to refer to the corresponding eigenvalue and use it in computations. Here is the issue: how do I call it?
I am afraid that $\lambda_{j}$ could be misleading, as it may not agree with the same $\lambda_{j}$ in the list $\lambda_{1}, \dotsc, \lambda_{k}$.
Any suggestion?
The notational convention I use here is to write $\lambda_1, \dots \lambda_d$ for the eigenvalues considered with multiplicity. Then $e_1, \dots e_d$ are the corresponding eigenvectors. This makes all the notation pretty consistent as long as you keep in mind that the $\lambda_i$ aren't necessarily distinct.