0
votes
1answer
21 views

SO(n) is parallelizable

Prove that $SO(n)$ is parallelizable. How would I go about showing this? My supervisor could not help me with this problem, and I am stumped.
0
votes
1answer
50 views

Centralizer of $SO(n)$

Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$? For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where ...
3
votes
0answers
59 views

Is there a name for the group of complex matrices with unimodular determinant?

Does the group $$ G = \left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\} $$ have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$. ...
4
votes
1answer
58 views

Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?

This might be a dumb question; I know only enough group theory to be able to ask dumb questions. Ken W. Smith has pointed out that one way to get intuition about the determinant is to observe that it ...
0
votes
1answer
43 views

homework: proof regarding skew-symmetric nondegenerate bilinear form

I have this proof for a homework exercise, which I can't figure out how to solve.. Let $V$ be a vector space over $F$ and let $B$ be a skew-symmetric or symmetric nondegenerate bilinear form on $V$. ...
1
vote
1answer
25 views

determinating the signature of a bilinear form

As as part of my course in Lie groups, I need some help to determine the signature of the form $B(x,y) = $$ \sum_{i=1}^{n}\ x_iy_{n+1-i}$ More than anything, I would like to understand how to ...
0
votes
1answer
46 views

Vector Space of Lie Algebra

Lie algebra $ \mathfrak{g} $ for a Lie group $ \mathcal{G}$ is closed under commutation. Also, the elements of Lie Algebra form a Linear Vector Space(LVS). Firstly, when is it allowed to define an ...
2
votes
2answers
52 views

Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
0
votes
0answers
20 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
0
votes
0answers
59 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
2
votes
0answers
25 views

Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper ...
2
votes
1answer
24 views

Questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$.

I have some questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in GL_2(\mathbb{F}_q)$ has eigenvalues which ...
1
vote
0answers
40 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
1
vote
2answers
69 views

Quaternion representation of rotations

How would one show that 1/3 turns correspond to the eight antipodal pairs among the 16 quaternions : $$ \pm \frac{1}{2} \pm \frac{i}{2} \pm \frac{j}{2} \pm \frac{k}{2} $$ knowing that the rotation ...
0
votes
0answers
51 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
10
votes
2answers
133 views

Is $SO_n({\mathbb R})$ a divisible group?

The title says it all ... Formally, if $SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$ and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is ...
0
votes
1answer
29 views

Matrix multiplication in $SO(3)$ that fixes row

I want to find all matrices $G \in SO(3)$ that do not change the first row of elements in $SO(3)$ when right multiplying by $G$, i.e. $$ \{ G \in SO(3): \forall A \in SO(3) \quad A = \begin{pmatrix} ...
3
votes
3answers
82 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
2
votes
2answers
70 views

How to show that exp is a diffeomorphism between symmetric reals and positiv definite matrices?

I am looking for an easy proof of the fact that the exponential function is a diffeomorphism between the finite dimensional vector space of symmetric real nxn-matrices and the open subset of positive ...
1
vote
1answer
65 views

A general element of U(2)

The group U(2) is the group of all $2\times 2$ matrices such that $U^\dagger U=I$. Evidently it has $4$ real parameters, and can be represented as: $$ U(2) = \{\begin{bmatrix}a&b\\ 0&d ...
4
votes
1answer
91 views

A Neat Rotation Matrix Identity

Let $\mathbf{R}_i$ be $N$ rotation matrices that represent a rotation around axes $\mathbf{\omega}_i$ by an angle $|\mathbf{\omega}_i|$. Now say we know that the product of these matrices is unity, ...
0
votes
1answer
45 views

Structural Question about Elements of $SO(2n)$.

Let $R_\theta$ denote the appropriate element of $SO(2)$, as is conventional. Let $n$ be a positive integer, and consider $D = \text{diag}(R_{\theta_1}, \dots, R_{\theta_n}) \in SO(2n)$. For ...
0
votes
1answer
25 views

Cofactors and conjugates of $SU(3)$.

I was playing around with some equations and noticed the following: Let $A$ be an element of $SU(3)$ with components $A_{ij}$. If $C_{ij}$ is the $(i,j)$ cofactor of $A$ then $C_{ij} = ...
3
votes
1answer
55 views

Is it true that stabilizer in $O(n)$ of a rank $k$ matrix is isomorphic to $O(n-k)$?

Let $X\in M_{n, k}(\mathbb R)$ such that $\textrm{rank}(X)=k$ and,$$O(n)_X:=\{A\in O(n): AX=X\}.$$ Notice $O(n)_X$ is a subgroup of $O(n)$. Is it true that $O(n-k)\cong O(n)_X$? Here $O(n)=\{A\in ...
3
votes
1answer
121 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
0
votes
0answers
23 views

Differences of skew symmetric matrices

Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on ...
1
vote
1answer
40 views

What 1D $\mathbb{C}$-Subspaces are Stabilized by Elements of a Specific 2-Torus in $SO(7)$?

Consider the 2-torus $T \subset SO(7)$ defined by $T = \left\{ \mathrm{diag}(R_{\theta_1}, R_{\theta_2}, R_{-(\theta_1 + \theta_2)}, 1) \mid \theta_1, \theta_2 \in \mathbb{R} \right\}$, where ...
3
votes
1answer
65 views

How to determine all conjugacy classes in the complex orthogonal group $O(n,\mathbb{C})$ of finite order $k$

I was wondering how one would go about determining the conjugacy classes of the complex orthogonal group $O(n,\mathbb{C})$ of some finite order $k$. That is, if $[A]$ is the conjugacy class of $A\in ...
3
votes
2answers
63 views

action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
2
votes
1answer
74 views

Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra

Consider following Lie Group: $$ \text{Sp}(2n,\mathbb{C})=\{g\in\text{Mat}_{2n}(\mathbb{C})\mid J=g^TJg\}\quad\ where\quad J=\begin{pmatrix} 0 & 1_n \\ -1_n & 0 \end{pmatrix} $$ And the ...
0
votes
1answer
40 views

Range of the exponential map

I need to build a matrix $A\in Gl^+_n(\mathbb{R})$ ($det(A)>0$) that is not an exponent, i.e. there is no $B\in Mat_n(\mathbb{R})$ such that $A=exp(B)$. Could you give me a hint to note some ...
3
votes
1answer
57 views

How to show $SL_{n}(\mathbb{R})=\bigsqcup_{w\in W}LwU$ where L (or U) are lower(or upper) triangular matrix?

I'd like to ask a homework problem that causes me many troubles for days. The problem is like below : Let W denote the subgroup of permutation matrices in $SL_{n}(\mathbb{R})$. Show the following ...
3
votes
3answers
105 views

Show that $\exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R)$ is not surjective

It is well known that for $n=2$, this holds. The polar decomposition provides the topology of $\operatorname{SL}(n,\mathbb R)$ as the product of symmetric matrices and orthogonal matrices, which can ...
1
vote
1answer
43 views

Why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?

Let $G=SO(n,R)$ be a Lie group and $\mathbb{g}$ its lie algebra. Take $X\in \mathbb{g}$. Then why is the coadjoint orbit passing through $X$ determined by the spectrum of $X$?
3
votes
1answer
107 views

Finding the Lie Algebra of a Lie Group

I am having a hard time finding the set of all $X \in M(n, \mathbb{R})$ such that $e^{tX^T}Be^{tX} = B$ for all $t \in \mathbb{R}$. where $b$ is any matrix in $M(n, \mathbb{R})$.
4
votes
1answer
92 views

orthogonal group of a quadratic vector space

I am reading about the orthogonal group $O(V)$ of a real finite dimensional quadratic vector space $(V,Q)$ with $Q$ nondegenerate. By definition $$O(V)=\{f:V\mapsto V |\quad Q(f(v))=Q(v) \quad \forall ...
6
votes
2answers
531 views

Two Definitions of the Special Orthogonal Lie Algebra

I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other. If we begin ...
1
vote
1answer
170 views

Determining which maps are isomorphisms

1) Let $G=G'=\{(a,b)\mid a,b \in \mathbb{R}, a, b \ne 0\}$ with group operation $(a_1,b_1)(a_2,b_2)=(a_1a_2,b_1b_2)$. Let $\phi (a,b) = (b^{-1}, ab^2)$. My solution: 1-1: Suppose ...
6
votes
1answer
180 views

Connectedness of the orthogonal subgroup $O^+_+(k,l)$

Let $O(k,l)$ be the orthogonal group associated to the quadratic form $q$ on $\mathbb{R}^{k+l}$ with signature $(k,l)$. Let $O^+_+(k,l)$ be the connected component of the identity, i.e. the connected ...
3
votes
1answer
99 views

Question about special orthogonal Lie group construction

Working through homework and I run into this problem: Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a ...
5
votes
1answer
315 views

The Gram-Schmidt process is a deformation retraction

Consider the Gram-Schmidt process $r : GL(n) \rightarrow O(n)$ that sends invertible matrices to orthogonal matrices. I need to show this is a deformation retraction and, by restrictions of $r$, ...
2
votes
1answer
85 views

Exponent of a matrix and commutative property

A question given at lie group theory course: $P$ and $Q$ are $n \times n$ matrices, and some $k \in \mathbb{R}$. Assume that $$\displaystyle\forall_{x,y \in \mathbb{R}} e^{xP}e^{yQ} = ...
0
votes
1answer
157 views

$3\times 3$ symmetric matrix with signature $(2,1)$

I need to show the set of $3\times 3$ real symmetric matrices with signature $(2,1)$ is an open connected subset in the usual topology of $\mathbb{R}^6$. To show connectedness I did like the ...
2
votes
1answer
36 views

How can I show that $ASL_n(F)$ is acting 2-transitively?

One of my friends asked me to ask this question here. This is a question from his last exam: Let $$ASL_n(F)=\{T_{A,v}:V_n(F)\to V_n(F)\mid\exists A\in SL_n(F), \exists v\in V_n(F), ...
0
votes
1answer
48 views

preserves eigen spaces?

"Let $H_0=\begin{pmatrix}i&0\\0&-i\end{pmatrix}$, suppose $A\in SU(2)$ commutes with $H_0$, it must preserves each eigen spaces for $H_0$, eigen spaces for $H_0$ are just $\mathbb{C}e_1$ and ...
6
votes
2answers
240 views

Why is the name general “linear” group?

Well, I just want to know if is there any significance of the term "linear" in the of name "General Linear Group" - for example, $\text{GL}_ n(\mathbb{R})$?
5
votes
1answer
596 views

Non surjectivity of the exponential map to GL(2,R)

I was asked to show that the exponential map $\exp: \mathfrak{g} \mapsto G$ is not surjective by proving that the matrix $\left(\matrix{-1 & 0 \\ 0 & -2}\right)\in \text{GL}(2,\mathbb{R})$ ...
2
votes
1answer
679 views

How to evaluate the derivatives of matrix inverse?

Cliff Taubes wrote in his differential geometry book that: We now calculate the directional derivatives of the map $$M\rightarrow M^{-1}$$ Let $\alpha\in M(n,\mathbb{R})$ denote any given matrix. ...
4
votes
2answers
143 views

If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?

Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$. Let $A$ be some ...