# Questions tagged [tensor]

The tensor tag has no usage guidance.

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### What are examples of "perfect tensors"?

A "perfect tensor" is defined on the nLab very abstractly as "its tensor/hom-adjuncts $V^{\otimes k} \to V^{\otimes n - k}$ for $k \le n/2$ are isometries". The only example I'm ...

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### Local diagonalisation of a degenerated 2d metric tensor

Consider a smooth 2d-manifold $M$ and let $g$ be a smooth $(0,2)$-tensor satisfying $rk(g)\geq1$ everywhere. Obviously if $rk(g)=2$ at a point $p\in M$ then $g$ is locally diagonalisable (i.e. there ...

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### What resource do Markov and Shi mean when they estimate tensor contraction complexity?

Markov and Shi in their paper Simulating quantum computation by contracting tensor networks define the contraction complexity as follows (page 10):
The complexity of π is the maximum degree of a ...

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### Decomposition of tensors

It is well known that every traceless, symmetric $2$-tensor can be decomposed uniquely into a Lie derivative part and a Codazzi part. Is there an analog for totally symmetric $k$-tensors?

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### Tucker decompositions over arbitrary fields

Given an $n$-mode tensor $\mathcal{T}\in\mathbb{R}^{d_1\times\dotsb\times d_n}$, there exists a Tucker decomposition of $\mathcal T$ of the form
$$\mathcal{T} = \mathcal{X}\times_1 W_1\times_2\dotsb\...

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### Is there a geometric intepretation of the trace of tensor on a Riemannian manifold?

For a long time I thought the trace for matrices were just an elementary function with nice properties. But it is much more than that and really should be think of as a geometrical object (see ...

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### How is this product of tensors defined?

I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...

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### Multi-variable expansion of $e^{u^T X}$ as a power series in terms of tensors [closed]

I have the following question related to multivariable moment generating functions. Given two vectors $u,X\in\mathbb{R}^d$, I want to characterize the quantity $e^{u^T X}$ in terms of "powers&...

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### The Ricci curvature is bounded below by scalar curvature

So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...

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### A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"

Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...

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### Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?

Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...

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### Proving the graded structure of the tensor algebra from only the universal property

When the tensor algebra is presented, it is usually constructed as the direct sum of all tensor powers of the space. By this construction, the graded structure of the tensor algebra is easy to prove. ...

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### Solving nonlinear differential multi-variable equation with block-matrices

Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...

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### Construct a 4th-order tensor with matricization ranks $r$ that is not rank $r$

I ask you for this possibly not so simple task:
Explicitly construct a 4th-order tensor $A \in \mathbb{C}^{n_1 \times \ldots \times n_4}$ that does not have (border) rank $r$, but for which each ...

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### Adjacency matrix/tensor operations for graph sequences?

Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise.
Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...

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### Derivative of eigenpair with respect to matrix

Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition
$$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$
where $\lambda_i$, $n_i$ and $\...

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### References Request: Bach tensor

Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...

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### Pseudo-tensor- and tensor-densities: Sections of what bundle?

Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...

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### Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$

I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...

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### Generalization of a standard algebraic group theory result for a tensor problem

$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...

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### Eigenvalues of large symmetric random tensors

I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting ...

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### Spectral theorem for symmetric real tensors

Is there a definition of eigenvalues that allows to use a spectral theorem?
Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...

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### Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3?
Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...

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### Can the eigenvalues of a real symmetric tensor be complex?

Let $T$ be a fully symmetric tensor of rank $3$ and size $N$.
Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that:
\begin{equation}
\sum_{jk}^...

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### Mean Gaussian curvature from non-unit vector

Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak.
Let us suppose that $\lambda_{h|}^i$
is not a unit vector and therefore, the mean curvature $M_h$ in
this ...

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### Quaternions as eigenvalues of rank 3 tensors

Let us consider a matrix $M^{(a)}$ of size $N \times N$, having $N$ eigenvalues $\lambda_i \in \mathbb{C}$.
Considering a rank-3 tensor, we can informally think of it as a sequence of $N$ matrices $M^{...

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### What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?

An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I ...

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### A different notion of a decomposable symmetric tensor (besides Veronese)

$\DeclareMathOperator{\complex}{\mathbb{C}}$
Let $\bigvee^m(\complex^n)\subseteq (\complex^n)^{\otimes m}$ denote the space of symmetric tensors, i.e. the set of $x \in (\complex^n)^{\otimes m}$ that ...

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### Christoffel symbols as the expansion coefficients of covariant/contravariant derivatives

Page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors state that Christoffel symbols of the second kind are expansions of $\frac{\partial {\bf e}_j}{...

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### What is the physical meaning of torsion

The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-...

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### Why some operations on tensors don't give a tensor? [closed]

I asked the following question on math.stackexchange but no one seemed to have an authorative answer so I'm posting here hoping that experts will see it.
The gradient is a tensor $\nabla f:\mathbf{V} \...

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### Best approximation with tensors of rank $\ge2$

Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ ...

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### Exactness of injective tensor products

For (algebraic) tensor products, it is well-known that the functor $A\otimes_R \cdot:Mod_R\rightarrow Mod_R$ is only (left-) exact when $A$ is a flat $R$-module. In particular, all vector spaces are ...

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### Analytical decomposed form of a specific traceless symmetric tensor

Assume an m-way tensor $\mathcal{Z}$.
$\mathcal{Z}_{p_1 p_2 ... p_m} = 0$ if any different indices match
and $\mathcal{Z}_{p_1 p_2 ... p_m} = 1$ otherwise.
It is a symmetric tensor. Now if it is 2-...

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### Estimates for tensors using local coordinates

Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...

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### Compute Frobenius inner product of two tensor-trains in terms of tensor contractions

Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...

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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
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...

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### 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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### 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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### 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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### 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 ...

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### 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)\...

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### 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 ...

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### 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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### 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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### 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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### 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 \...

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### 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} (\...

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### 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 extremely ...

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### 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 ...