# Terminology for different types of eigenvalue degeneracy?

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?

• 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. – Christoph Feb 2 at 23:32
• This question introduces "algebraic multiplicity" versus "geometric multiplicity". – Roy Simpson Feb 5 at 19:25