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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.

23 questions
7
votes
1 answer

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 $G$ is unipotent if its adjoint action $x \mapsto…
user512723
6
votes
3 answers

$n$-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 that for every nilpotent $N \in Mat_{3, 3}…
user40276
  • 5,543
4
votes
2 answers

Center and Commutator of Profinite Groups

I am working in a group called $$\mathrm{UI}_n=\{A\in \mathrm{GL}_n(\mathbb{F}_q[[t]]) \mid A_{ij}(0) \neq 0 \text{ implies } i\leq j, i=j \text{ implies } A_{ij}(0)=1\}$$ Which can be thought of as the inverse limit indexed by k of…
Mattan Feldman
  • 324
4
votes
1 answer

Unipotent elements vs. unipotent linear transformations

Let $n\ge 2$ and $A\in GL(n,\mathbb R)$ be a matrix and define a linear transformation on $M(n,\mathbb R)$ as follows: $$f_A: M(n,\mathbb R) \to M(n,\mathbb R), B \to ABA^{-1}. $$ Suppose $f_A$ is a unipotent linear transformation on $M(n,\mathbb R)…
No One
  • 8,465
3
votes
1 answer

Unipotent matrices are conjugate to their inverses

An element $a$ in a ring $R$ is called unipotent if $(1-a)^k=0$ for some positive integer $k$. Let $F$ be a field and $n$ a positive integer. It is not difficult to see that each unipotent element in the ring of matrices over $F$ is of forms an…
Tran Nam Son
  • 813
2
votes
0 answers

Specific matrix decomposition

Suppose we have a symmetric matrix $A$. Is it possible to find unipotent matrix $U$ and diagonal matrix $D$ such that $U A U^T = D$?
Betydlig
  • 425
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votes
0 answers

Characteristic subgroups of the group of unipotent matrices

Let $\mathbf{U}(n,p)$ be the group of $n\times n$ upper triangular matrices over $\mathbf{F}_p$, with $1$ all over the diagonal. Let me number the paris $(i,j)$ , $1\leq i
Gaussian
  • 402
2
votes
2 answers

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 unipotent if and only if there is $k\in…
Andrew
  • 713
2
votes
0 answers

$\mathbb{F}_q$-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 the finite field…
kneidell
  • 2,520
2
votes
2 answers

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 $U \in \mbox{Mat} (3,3,\mathbb{R})$ and a diagonal matrix $D \in \mbox{Mat}…
Taufi
  • 1,113
1
vote
0 answers

Unipotent closure in classical groups

Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then Ratner's orbit closure theorem states that for…
Mathew
  • 33
1
vote
1 answer

$A$ is unipotent iff all of its eigenvalues are 1

Note: A is an element of the ring of n-by-n matrices with complex entries. I know how to prove that $A$ is nilpotent iff all its eigenvalues are 0, but I'm not sure how to prove that $A$ is unipotent iff all of its eigenvalues are 1. For the right…
beginner
  • 1,908
1
vote
0 answers

On characters of unipotent groups

I was thinking about characters of unipotent groups, this is my question. Let's define a unipotent group as an algebraic linear group formed by unipotent elements, where an element is called unipotent if it acts unipotently on some regular…
wood
  • 311
1
vote
1 answer

Unipotent matrix with all block matrices having non-zero determinant

Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i.e. $M$ is a unipotent matrix). Write $M=\begin{bmatrix} A_1&A_2\\ A_3&A_4 \end{bmatrix}, $ where each $A_i$ is a $2$ by $2$ sumbatrix of $M$. Let $a_i =…
ghc1997
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1
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0 answers

Product law on complex number via unipotent group

It is a well-known fact that $\mathbb C$ has the group structure with respect to the sum. Identifying $\mathbb C$ with the subgroup $$ U_a=\left\{ \begin{pmatrix} 1 & 0 \\ c & 1 \end{pmatrix}: c \in \mathbb C \right\} \subset…
Bobech
  • 442
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