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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?

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  • $\begingroup$ I don't know about a specific terminology for this, but the partition $(2,1)$ in your example is the shape of the Young diagram (or dot diagram) associated to the generalized eigenspace when computing the Jordan canonical form. $\endgroup$ – Christoph Feb 2 at 23:32
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    $\begingroup$ This question introduces "algebraic multiplicity" versus "geometric multiplicity". $\endgroup$ – Roy Simpson Feb 5 at 19:25

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