All Questions
10 questions
3
votes
0
answers
46
views
Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size
I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\...
- 433
1
vote
1
answer
319
views
Are there any applications of linear algebra over the complex numbers, where the role of complex conjugation is replaced with the trivial involution?
The complex inner product $\langle u, v \rangle$ is a special case of a sesquilinear form over a field. Its definition is $\langle u, v \rangle = \sum_{i} u_i \overline{v_i}$. There is clearly the ...
- 4,943
6
votes
3
answers
327
views
$SO(m+1)$-equivariant maps from $S^m$ to $S^m$
Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$.
Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$.
Question 1: Is F the ...
- 28k
4
votes
1
answer
696
views
$\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?
Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
- 1,540
4
votes
2
answers
758
views
Riemannian metric of hyperbolic plane
I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
- 8,597
0
votes
0
answers
53
views
Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action
I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...
- 2,099
6
votes
2
answers
426
views
Approximating the action of the U(N) exponential map
Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group:
$$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$
Here, $H$ is an NxN skew-Hermitian matrix (for very ...
- 20.2k
7
votes
2
answers
315
views
Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices
I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...
- 6,206
3
votes
1
answer
367
views
What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?
If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
- 108k
3
votes
1
answer
494
views
A question on Grassmannian
Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...
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