Let $T$ be a linear operator on a finite-dimensional complex vector space $V$, and let $\lambda$ be an eigenvalue of $T$ with multiplicity $m$ (defined as the dimension of the subspace of $V$ spanned by generalized eigenvectors of $\lambda$).
At least one of the generalized eigenvectors of $V$ must be a “proper” eigenvector, but it may turn out that there are more than one (linearly independent) proper eigenvectors. For example, the operator defined by the matrix
\begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{bmatrix}
has one eigenvalue, $0$, with multiplicity $3$. But it also seems noteworthy that it has two proper eigenvectors that are linearly independent: $(1,0,0)$ and $(0,1,0)$.
Is there good terminology to express this fact? For example, could I say that the eigenvalue of $0$ for the above operator is “$(2,1)$-degenate,” meaning it has multiplicity $2+1=3$, where there are $2$ eigenvectors and $1$ generalized eigenvector, all of which are linearly independent?