Questions tagged [tensor]
The tensor tag has no summary.
163 questions
0
votes
0
answers
141
views
On tensor product and rank
I am getting confused by the tensor product. I would appreciate some basic insight.
I consider $M_2\otimes M_3$. (Here $M_n$ denotes the complex $n\times n$ matrices.) The dimension of this space is 4*...
- 109
1
vote
0
answers
118
views
Is there a generalization of Gamas's theorem that works for any finite permutation group?
Let $n$ be a positive integer and let $G$ be a subgroup of $S_n$. Let $\rho$ be an irreducible finite-dimensional representation of $G$ with character $\chi_\rho$.
Let
$$\pi_\rho = \frac{\chi_\rho(e)}{...
- 5,357
3
votes
0
answers
106
views
Duality for "sum of squares" tensors
Let $T$ be a symmetric tensor in $\left( \mathbb{R}^d \right)^{\otimes n}$ such that the polynomial $$\sum_{i_1,i_2, \ldots i_n}T_{i_1i_2 \ldots i_n} x_{i_1}^2 x_{i_2}^2 \ldots x_{i_n}^2$$ is a sum ...
3
votes
0
answers
224
views
Decomposition of tensor invariant under $SO(n)$
I am working with spherical harmonics on $S^n$ where orthonormal elements of $L^2(S^n)$ will be denoted $h_\alpha (x)$ i.e.:
$$
\int\mathrm{d}\Omega\;h_\alpha(x)h_\beta(x)=\delta_{\alpha\beta}
$$
I ...
- 31
0
votes
1
answer
339
views
Prove that a matrix is almost surely full rank
[Cross-posted from MS after 8 days without reply.] I have a real matrix $R_{ij} \in \mathbb{R}^{n \times m}$ whose entries are sampled iid from an absolutely continuous distribution $D$; a fixed real ...
- 155
1
vote
0
answers
86
views
Rank of tensors with a $G-$action
I've come across this question while trying to prove a divisibility criterion for the rank of symmetric tensors.
The problem can be stated in much more generality, and can be generalized as follows:
...
- 1,353
0
votes
0
answers
79
views
Least-square distance between an array of quadratic forms and a given positive vector
Suppose we are given a list of $N$ positive definite quadratic forms $X^TQ_k X$ (where $k\in[1,N]$ and
$Q_k\in\mathbb{R}^{p\times p}$ $\forall k$), and a positive vector $V$ of same length $N$ i.e. $V=...
- 1
5
votes
1
answer
244
views
Invariant theory for unitary groups $\mathcal{U}(n)$
I'm trying to understand the invariant theory of the unitary groups $\mathcal{U}(n)$ on tensor powers of their standard representations $V^{\otimes p} \otimes (V^*)^{\otimes q}$. Let $\mathcal{U}(n)$ ...
- 1,184
3
votes
1
answer
666
views
Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
user199422
0
votes
1
answer
225
views
Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 45
1
vote
0
answers
162
views
Space of all orthogonal partially complex $2\times2\times2$ tensors
I am trying to understand the space of all orthogonal tensors, I asked a more general version of this question here but with no solution yet found. I want to look at the simplest case first, namely a $...
- 101
4
votes
0
answers
168
views
Space of all orthogonal $2\times2\times2$ tensors
I am trying to understand the space of all orthogonal tensors, a question asked here before but with no real solution yet found. The solutions for order-$2$ tensors are clear so thus the simplest case ...
- 101
2
votes
0
answers
80
views
Adjoint to "strict twocategory of strict twofunctors"
Let C be the category of strict twofunctors, featuring the addition of a Grothendieck universe. Strict twocategories are categories enriched over the category of categories.
C has an internal hom ...
user531303
1
vote
0
answers
177
views
Can numerical differentiation be applied to tensor derivatives?
I know that for a 1D function, I can calculate the numerical derivative at every point, $\DeclareMathOperator{\d}{d\!} (x_1,y_1)$, with $\d y/\d x$ where $\d y = y_2 - y_0$ and $\d x = x_2 - x_0$. If ...
- 19
2
votes
1
answer
248
views
Combination of simple tensors - II
This is a follow-up question to Combination of simple tensors.
I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
