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Questions tagged [unipotent-matrices]

A square matrix $A$ is unipotent if $A-I$, where $I$ is an identity matrix, is nilpotent.

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2answers
47 views

Unipotent matrix similar to an upper-triangular matrix

"Any unipotent matrix is similar to an upper-triangular matrix with 1's on the diagonal"... This is usually alleged, but I have no idea how to demonstrate that, starting with the definition : $A$ is ...
1
vote
1answer
43 views

Unipotent action on flag variety

I am reading https://www.sciencedirect.com/science/article/pii/S138572587680008X and must be missing something obvious. The premise is that we are considering the fixed point set of a unipotent ...
2
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0answers
37 views

$\mathbb{F}_q$-rational elements in unipotent classes of simple algebraic group in positive characteristic

Sorry in advance if this question is trivial or trivially false. I haven't managed to find a satisfactory proof (or reference of one), or a counterexample for it. Let $k$ be the algebraic closure of ...
3
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0answers
125 views

Unipotent elements in a Lie group

In a matrix Lie group $G$, we say that $g\in G$ is unipotent if $$(g-I)^n=0 $$ for some $n\in \mathbb{N}.$ I read in a Tao's article, that More generally, we say that an element g of a Lie group ...
1
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1answer
35 views

Unipotent elements of $\mathrm{GL}_n(\mathbb{F}_p)$ have order $p^k$, $k\in \mathbb{Z^+}$

This is one part of a larger question I'm trying to answer, but I'm stuck on this: If $u \in \mathrm{GL}_n(\mathbb{F}_p)$, the general linear group of degree $n$ over $\mathbb{F}_p$, and $u$ is ...
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1answer
27 views

Finite algebraic automorhisms of unipotent group.

Let U be a subgroup of unipotents matrices(A - E is nilpotent). U is affine algebraic group. How to describe finite($\phi^n = id$) algebraic automorphisms $\phi$ of U such that $U^{\phi}$ is finite? ...
5
votes
3answers
173 views

$n$-th root of $3 \times 3$ invertible matrix

Yo, I couldn't solve this exercise after thinking for a while. For every $A \in GL_{3} (\mathbb{C})$ and $n$, there's a $B \in Mat_{3, 3}(\mathbb{C})$ such that $B^n = A$ The previous exercise was ...
2
votes
2answers
128 views

Matrix decomposition in unipotent matrices

Consider the positive definite and symmetric matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 6 & -1 \\ 0 & -1 & 1 \end{pmatrix}$$ Find a decomposition with unipotent $U ...