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Results tagged with linear-algebra
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user 142627
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
6
votes
2
answers
801
views
Cyclic vectors in irreducible representations of simple Lie algebras
Is there a notion of "cyclic element" in a simple Lie algebra? In particular, is it independent of the irreducible representation chosen?
Explanation.
An endomorphism A is called cyclic if there …
- 617
4
votes
2
answers
352
views
Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the cen …
- 617
0
votes
Accepted
Double centralizer in special linear algebra
I just found a counter-example for $\mathfrak{sl}_3$.
Take $A= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ and
$B = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bm …
- 617