Notation for eigenvalue corresponding to the $i$th eigenvector

Question

Given some $n \times n$ matrix with $m\le n$ distinct eigenvalues, I usually see the spectrum denoted as distinct eigenvalues $ \lambda=\{ \lambda_1, \ldots,\lambda_m \} $ with multiplicities $ \mu=\{ \mu_1, \ldots,\mu_m \} $. For example, given $$ A=\begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & 4 & 0 & 0\\ 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ would denote $\sigma(A)$ as $ \lambda=\{5,4,1\} $ with multiplicities $ \mu=\{1,2,1\}$.

How is the eigenvalue corresponding to the $i$th eigenvector usually denoted? i.e., how do I denote both of:

1. the third (distinct) eigenvalue of $A$ is $1$?
2. the third (including non-distinct) eigenvalue of $A$ is $4$?

I am currently thinking:
$\lambda^{(i)}$ refers to the $i$th (distinct) eigenvalue such that $\lambda^{(1)}>\dots>\lambda^{(m)}$
$\lambda_i$ refers to the $i$th (including non-distinct) eigenvalue such that $\lambda_1\ge\dots\ge\lambda_n$
According to this schema $\lambda^{(3)}=1$ and $\lambda_3=4$.
Is this reasonable?

Context of question:
I am writing an analysis of an algorithm which utilises eigenprojections. I frequently need to denote both the $i$th distinct eigenvalue and the eigenvalue corresponding to the $i$th eigenvector.

Thanks in advance for your patience.

Answer 1

I would say $\lambda_{i}$ the $i$th eigenvalue, which has $E_{i}$ as eigenspace.