All Questions
Tagged with lie-groups linear-algebra
116 questions
4
votes
2
answers
496
views
Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group?
I need to answer (affirmatively, I hope) the following question:
In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset
...
5
votes
1
answer
404
views
determining symplecticity (if that's a word)
Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the ...
2
votes
1
answer
297
views
equations over (some) lie groups
To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the ...
2
votes
1
answer
232
views
An innocent looking subgroup of $U(n)$
Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...
6
votes
3
answers
482
views
Linear subspaces in cones over orthogonal groups
Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...
6
votes
0
answers
465
views
Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
13
votes
1
answer
732
views
What is the "positive part" of the unit ball in $M_n(R)$ ?
In ${\bf M}_n(\mathbb R)$, let us consider the usual operator norm
$$\|A\|=\sup\frac{\|Ax\|}{\|x\|},$$
where $\|x\|$ is the Euclidian norm.
The closed unit ball $B$ is the set of contractions (in the ...
6
votes
4
answers
1k
views
Polar decomposition for quaternionic matrices?
A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, ...
0
votes
0
answers
172
views
Generating Set for $O(V)$ over $\mathbb Z_2$
I am reading a claim that $O(V)$ — the orthogonal group associated with a finite-dimensional vector space $V$ over $\mathbb Z_2$ and a quadratic form $q$, i.e. the group of linear ...
3
votes
3
answers
461
views
Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
0
votes
1
answer
152
views
"locally" factoring subgroups of Lie groups
I'm not really a math person, and apologize if the question here is too simple. I've ended up with the following type of question for a few Lie groups, but state it for SO(n).
I start with a subgroup ...
5
votes
4
answers
3k
views
Parametrization of O(3)
Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?
0
votes
0
answers
608
views
Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
13
votes
2
answers
3k
views
Left and right eigenvalues
A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...
-1
votes
2
answers
806
views
The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
7
votes
2
answers
1k
views
Abelianization of Lie groups
If G is a group, its abelianization is the abelian group A and the map G → A such that any map G → B with B abelian factors through A. Abelianization is a functor, and in general a very ...