# Questions tagged [tensor]

The tensor tag has no usage guidance.

96
questions

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### Characterization of all-orthogonal tensors

In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...

**3**

votes

**1**answer

80 views

### Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...

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38 views

### Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...

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14 views

### Show that a tensor-train is contained in a recursive sequence of subspaces

Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...

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votes

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30 views

### How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?

Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...

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votes

**1**answer

45 views

### Cores in the tensor-train decomposition

Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...

**1**

vote

**1**answer

96 views

### Symmetric tensor components

EDIT: I thought on rephrasing the question in another way:
I have been working recently with a tensor that satisfies
$A_{ijkl}=A_{i+b,j+b,k+b,l+b}$ $\forall$ $i,j,k,l$ $\in$ Z
$$dist(i,j,k,l)\...

**4**

votes

**1**answer

246 views

### Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor

I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues ...

**8**

votes

**1**answer

139 views

### Question on Nash's paper on $C^1$ isometric immersions: Why approximating the error tensor $\delta$?

I am trying to go through the classical paper by Nash on the existance of $C^1$ isometric immersion of a Riemannian manifold $(M,g)$ (here is the Jstor link: https://www.jstor.org/stable/1969840?seq=1#...

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20 views

### Spectral resolutions of a second-order orthogonal tensor

Page 37 of Continuum mechanics by C. S. Jog lists the following formulae as the "spectral resolutions" of an orthogonal tensor $\bf R$ as having the eigenvectors ${\bf e} , \, {\bf n} , \, {\bf \hat{n}...

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69 views

### Most tensor subspaces of low dimension have rank-1 defining equations

Let $V_1,\ldots , V_k$ be vector spaces of dimensions $n_1,\ldots , n_k$ over a field of characteristic zero.
Consider the rational map
$
\newcommand{\PP}{\mathbb{P}}
\newcommand{\bs}{\boldsymbol}
\...

**2**

votes

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78 views

### Naive generalization of determinant from matrices to higher rank tensors

Recall that using the Levi-Cevita symbol the determinant can be written as
$$\operatorname{det} A=\frac{1}{n!}\epsilon_{i_1\dots i_n}\epsilon_{j_1\dots j_n}A_{i_1j_1}\dots A_{i_nj_n}$$
Some ...

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**0**answers

63 views

### For a manifold of positive curvature, can we lower bound the distance between unit normals?

Suppose $M \subset \mathbb R^d$ is a $C^2$ $(d-1)$-manifold. In particular I am interested in when $M$ is the set $\{x \in \mathbb R^d: f(x) \le 1\}$ for some $C^2$ function $f:\mathbb R^d \to \...

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127 views

### Given a specific tensor on $V^{\otimes d}$, which $GL(V)$ orbit does the tensor belong to?

Suppose we have a vector space $V$ over a field $K$, with basis vectors $\hat{\bf{e}}_k$, and suppose we define a tensor
$$
\Lambda = \lambda_{i_1, ..., i_d} (\hat{\bf{e}}_{i_1} \otimes ... \otimes \...

**0**

votes

**1**answer

78 views

### Trace of a finite hypercubic tensor

Is the trace of a finite hypercubic tensor defined?
Clearly, for the bidimensional case $n \times n$ the trace is defined as the sum of the elements on the main diagonal:
$$\operatorname {tr} (\...

**4**

votes

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73 views

### Are mixed discriminants and hyper-determinants the same thing?

Premise
Hello, I'm trying to learn more about hyperdeterminants, but as I'm sifting through the literature I'm finding more and more papers on Mixed discriminant and I find their definitions ...

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92 views

### Images and Kernels of tensor products of homomorphisms of modules

Let $\mathcal{H}$ be a Hopf algebra and let $M_1,M_2,N_1,N_2$ modules over $\mathcal{H}$. If $f:M_1\rightarrow N_1$ and $g:M_2\rightarrow N_2$ are homomorphisms of modules, then are the following ...

**2**

votes

**0**answers

46 views

### Rank-1 decomposability of symmetric tensors

My question is about rank-1 decomposability of symmetric tensors over the reals.
Let $v_1,\dots,v_n\in\mathbb{R}^d$ be vectors. Construct the object:
$$
V=\sum_{j=1}^n \underbrace{v_j\otimes v_j\...

**0**

votes

**1**answer

112 views

### Rank of matrices and secant varieties

Consider the Segre embedding $\mathbb{P}^n\times \mathbb{P}^n\rightarrow \mathbb{P}^N$, and let $S\subset\mathbb{P}^N$ be its image.
Then $rank(Z)\leq k$ implies that $Z\in Sec_k(S)$. Moreover if $Z\...

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36 views

### On symmetric tensors with same rank, different orders

Let $A,B$ be two symmetric tensors of same rank $m$; and orders $k$ and $\ell$, respectively. In particular, assume that $A,B$ admits the following structure: There exists $v_1,\dots,v_m\in\mathbb{R}^...

**4**

votes

**1**answer

218 views

### Symmetric tensor decomposition

Let $T$ be an order-$k$, rank-$m$ symmetric tensor, that is, $T=\sum_{j=1}^m v_j\otimes v_j \otimes \cdots \otimes v_j$, where the Segre outer product is taken $k$ times, with $v_j\in\mathbb{R}^d$ for ...

**0**

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**1**answer

192 views

### Sketching Frobenius norm of a tensor with a rank-1 random tensor

Let $A\in\mathbb{R}^{n^k}$ be a $k$-dimensional tensor with $n$ elements along each dimension. Moreover suppose $u_1,u_2,\dots,u_k\sim\text{Unif}(\pm1)^n$ are $n$ dimensional vectors with each of ...

**1**

vote

**1**answer

58 views

### Connection of the existence of Killing-Yano tensor and Killing tensor

Stephani states that in 4 dimensions a spacetime admits a non-reducible Killing-Yano tensor only if the Weyl tensor either is
of Petrov type D or vanishes. Does this imply that the spacetime also ...

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79 views

### Example Petrov Classification

I would like to calculate the Petrov type for a specific spacetime, unfortunately I am not able to find a step by step algorithm or example for the process, either using null-tetrad, nor calculating ...

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votes

**1**answer

168 views

### How to count the number of tensors over a finite field of tensor rank $r$?

For simplicity, work over $\mathbb F_2$ and only consider order-$3$ equilateral tensors. For $r\in\mathbb Z_{>0}$, how many tensors $\mathfrak{T}\in\mathsf{Ten}_{n}^{\otimes 3}(\mathbb F_2)$ are ...

**7**

votes

**1**answer

147 views

### Non-tensor-representable ultrafilters on $\omega$

If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\...

**3**

votes

**2**answers

117 views

### Rank of order-3 tensor with all slices being rank-1

If some tensor $T=(t_{ijk})$ has that all of its (2 dimensional) slices (along all 3 axes) are of rank-1, does it follow that the tensor is also rank-1? That is, can be written as
$$ t_{ijk}=a_i b_j ...

**2**

votes

**1**answer

347 views

### Eigenvalue and Eigenmatrix of a 3D Tensor - How to calculate it?

How to calculate easily the eigenmatrix of a 3D tensor.
I try immersing the tensor in a big matrix, in my case, the tensor is of nxnxn and I can build an n^2 x n^2 matrix that contains all the "...

**1**

vote

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276 views

### On a Riemannian manifold, calculate the metric from the distance [closed]

Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...

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286 views

### Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called
Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.
Carleman linearization is a technique used to embed a finite
...

**7**

votes

**1**answer

178 views

### Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...

**2**

votes

**1**answer

539 views

### Is there a generalization of eigenvalues and eigenvectors to tensors?

Two perhaps ill-posed or just silly questions:
Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...

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96 views

### Product of cotton-york tensor with ricci tensor

In the process of calculation of a problem in tensor analysis, I have encountered with an expression given by $$C^{ij}R_{ij}=0.$$
This is for a Riemannian manifold of dimension 3. Assuming the ...

**2**

votes

**2**answers

454 views

### A kind of “Curvature tensor” for higher dimensional tensors

I begin my question with a multilinear question then I will consider two local smooth analogies:
Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...

**5**

votes

**1**answer

391 views

### higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia)
If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...

**1**

vote

**1**answer

149 views

### Strassen-like algorithm for Hadamard product of $2\times 2$ matrices

Let $A=\pmatrix{a_1& a_2\\a_3&a_4}, B=\pmatrix{b_1& b_2\\b_3&b_4}$ be two matrices. Let $C$ be the Hadamard product of $A$ and $B$.
$$C=\pmatrix{c_1& c_2\\c_3&c_4}=\pmatrix{...

**5**

votes

**1**answer

92 views

### Looking for a tractable algorithm or formula for the determinant of a tensor

It is possible to define the determinant of a tensor.
We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map.
If $T:\{1,....,n\}^m\to \mathbb C$ ...

**1**

vote

**0**answers

49 views

### Bounds on the tensor and border rank ratios of tensor unfoldings?

Consider $T_1 \in \mathbb{C}^{d_1 \times \cdots \times d_k}$ as an order-$k$ tensor of the format $(d_1, \cdots , d_k)$. Now, let $T_2$ be an unfolding of $T_1$ that is obtained by merging $\mathbb{C}^...

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26 views

### Non Negative Tensor Tucker Decomposition Error Degradation

I have been working on iterative decomposition methods of tensors with non negativity constraints. I have noticed that $\textbf{N}$on negative $\textbf{T}$ensor $\textbf{F}$actorization "NTF" which is ...

**8**

votes

**1**answer

271 views

### Geometric Interpretation of Multiplication in Pure Cubic Number Fields and Beyond

I got interested in the question of possible geometric interpretations of the multiplication in algebraic number fields of degree $>2$ (with application to multiplication of units in the ring of ...

**3**

votes

**1**answer

150 views

### Multi-dimensional permanent

Is there a particularly natural / "correct" way of generalizing permanents to tensors? (I mean of course, 'square' tensors.) There seem to be very few resources on the matter. There needs to be a ...

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votes

**1**answer

156 views

### Vanishing of determinant of Cotton York tensor

suppose $\Omega \subset \mathbb{R}^3$ is a Riemannian manifold with $g= dx_1^2 + dx_2^2 + c(x_1,x_2,x_3)dx_3^2$. Is it true that $det(CY)=0$?
Thanks

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**1**answer

298 views

### Is a flattening rank a lower bound for the border rank?

Suppose $T \in V_1 \otimes \cdots \otimes V_k$ is a tensor, where each $V_i$ is a finite dimensional complex vector space. A $1$-flattening (or a flattening) is a realization of $T$ as a matrix in the ...

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68 views

### Finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry

The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/...

**2**

votes

**3**answers

444 views

### Tensor Field Decomposition in Space time

For vector fields in $R^3$ one knows that there exists a unique decomposition of vector fields in to solenoidal (divergence free) and potential parts. A generalization of this Theorem for symmetric ...

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**1**answer

186 views

### Relation between curl and tensor divergence

The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas.
Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\...

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votes

**2**answers

515 views

### Derivative of eigenvalues w.r.t. a tensor

$E$ is a real, positive-definitive 3x3 symmetric tensor (I am thinking about the strain tensor in solid mechanics). We perform eigendecomposition and get:
$$E_p=\sum_{i=1}^{3}λ_iN_i⊗N_i$$
into its ...

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63 views

### Doubt on the best low rank approximation of a symmetric tensor

I have a matrix $M\in\mathbb{R}^{n\times k}$, with $k<n$ whose columns $m_i$ are linearly independent.
So we have $M := [m_1|..|m_k]$.
From the columns of $M$ I can define the following matrix
$$
...

**4**

votes

**1**answer

190 views

### Tell me something about these “component tensor” TQFT's

I noticed that there is a class of TQFT's that exists for every dimension $n\geq1$. It's probably well-known because it's quite simple, but I'm looking for a standard name or a better way to think ...

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195 views

### Smoothness of a (given) global section of a vector bundle over G(2,6)

Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong $ det$(S^*)$, $S$ being the ...