Questions tagged [tensor-rank]

For questions about tensor-ranks.

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Why can we substitute $ V_{\mu \nu} $ to $ V_{\mu ; \nu} $ while inducing contracted Bianchi identity?

After $ ( A_\mu B_\nu )_{; \sigma ; \rho} - ( A_\mu B_\nu )_{; \rho ; \sigma} = A_\alpha B_\nu R^\alpha_{\mu \rho \sigma} + A_\mu B_\alpha R^\alpha_{\nu \rho \sigma} $ where $ A_\mu B_\nu $ is outer ...
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Did I perform the partial derivative correctly?

I have the following expression: $\gamma^s=\sum_{s'}(X^{ss'})^{-1}m^{s'}(C-\sigma\otimes I)\cdot\varepsilon$ Where $s$ is the number of slip systems in the material and can be any integer. For this ...
Jesse Feng's user avatar
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tensors of type $(q,r)$ contra-variant and co-variant

Are the positions of $q$ and $r$ correct in the formula $5.29$ taken from the book by Nakahara: Geomtery Topology and Physics ? I think, that $q$ denotes the covariant part and hence should be $$dx^{\...
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Interpretation of $(2,0)$-tensors

I am aware that $(0,2)$-tensors are things like an inner product, a differential $2$-form, and more generally a bilinear map. I am aware that $(1,1)$-tensors are just linear endomorphisms. But what ...
Eduardo de Lorenzo's user avatar
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connection between $x\otimes f$ and rank-one operators

Let $X$ be a Banach space.Then for every nonzero $x\in X$, and nonzero $f\in X^*$, I'm told that the tensor product $x\otimes f: X\rightarrow X$ defined by $(x\otimes f)y=f(y)\,x$ is a rank-one ...
OSCAR's user avatar
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Misunderstanding Einstein notation

I have an expression $$p_m p_j (\delta_{ij} r_m + \delta_{jm} r_i + \delta_{im} r_j)$$ So if we expand things out $$=(\delta_{ij} r_m p_m p_j+ \delta_{jm} r_i p_m p_j+ \delta_{im} r_j p_m p_j)$$ ...
Stephen 123's user avatar
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Show that $T_{ijk}V_k$ is a second rank tensor in 3D space

The task is show that $$T_{ijk}V_k$$ is a rank-2 tensor. Using $$T_{ijk} = a_{i\alpha}a_{j\beta}a_{k\gamma}T_{\alpha\beta\gamma}' $$ and $$V_{k} = a_{k\gamma}V_{\gamma}' $$ I arrive at $$T_{ijk}V_k = ...
Abe Jacob's user avatar
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Algebra equation for 3 rank tensor

Suppose I work in $4$ dimensions. I have an algebraic equation in the following form, which contains a 3 rank tensor $X ^{\alpha \lambda \mu }$ \begin{equation} X ^{\alpha \lambda \mu }\eta ^{\beta \...
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Understanding the Definition of a Rank-1 Tensor

A tensor is nothing but a multidimensional array. We can think of an $n-mode$ tensor as a structure whose each element has to be referred with the help of $n$ indices or $n$ axes. Now, while reading ...
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Reference request: Book on theory of rank of matrices and multi-linear operators!

Is there a reference out there that only focus on (different)rank of matrices(with all kind of entries: real, complex, integers) and connects then further to ranks of tensors and further with the ...
Sarthak's user avatar
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Prove a tensor identity

Prove the tensor identity: $$e_{ij}^2-\frac{1}{3}e_{kk}^{2}=(e_{ij}-\frac{1}{3}\delta_{ij}e_{kk})^{2}$$ This equality we used it to prove the conservation of energy equation although i don't see ...
Ivan Bravo's user avatar
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Higher order Tensor computations

I have a $D$-way tensor H of dimension $I \times I \times \dots \times I$ ($D$ times), that represent the coefficients of a polynomial. For better understanding, I provided an image of 3-way tensor ...
Neuling's user avatar
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How to represent a bivector/bi-covector or more complex tensors (of rank 1,2/2,1/2,2 or even higher) using numbers?

If I understand correctly, the rank 0,0 tensor is just a scalar, rank 0,1 tensor is a vector represented as a column, rank 1,0 is a covector represented as a row, rank 1,1 is a linear transformation ...
Join the party P.A.R.T.Y.'s user avatar
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Symmetric Trilinear map for proving Newton's method on self-concordance functions

The following claim is used for proving the quadratic convergence phase of Newton's method on self-concordance functions. I have a very long and non-intuitive proof using Lagrange Multipliers. I ...
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show that the mixed derivative of zero order tensor gives second order tensor [duplicate]

I am trying to prove that for a zero order tensor, $f$ the expression $\partial^2f \over \partial x_i \partial x_j$ is a second order tensor, yet struggle to do so. I'd appreciate your help.
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Tensorial Representation of a Complex Network (Questions on Tensors)

INTRODUCTION TO QUESTION 1 Some authors proposed a tensorial representation of complex networks (for both single layer networks and multilayer networks). One reference paper for this topic is this one:...
Ommo's user avatar
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How do you determine the rank of the differential of an immersion?

I'm trying to prove that some mapping of a manifold is also a manifold using the constant rank theorem but I don't really understand the meaning of rank in this context. As I understand it, the ...
Username_57's user avatar
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Does "$\Gamma_{k i j}=\Gamma^m{ }_{i j} g_{m k}$" mean tensor-contraction or multiplication?, i.e. $\Gamma_{k i j}=\Gamma^m{ }_{i j} \cdot g_{m k}$?

Does $\Gamma_{k i j}=\Gamma^m{ }_{i j} g_{m k}$ mean tensor-contraction or does it mean multiplication, i.e. $\Gamma_{k i j}=\Gamma^m{ }_{i j} \cdot g_{m k}$ An alternative definition, I cannot ...
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Can a rank 1 tensor represent the same information that a rank n tensor does?

So, I was studying quantum mechanics, and a question came up. In quantum mechanics, we work in a Hilbert space, and the basis that we choose can represent all the information that we can have in order ...
Lucas Sievers's user avatar
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What is the reasoning/need for writing tensor indices like $T^i{\,}_j^{{\,\,}k}$ instead of $T_{j}^{ik}$ (or, $A^i_{\,\,\,j}$ instead of $A^i_j$)?

In one of the comments to this previous question I wanted to know why one of the users that gave an answer kept writing connection terms as ${\Gamma}^{\,\,\,l}_{h\,\,k}$. Until I read that answer I ...
FutureCop's user avatar
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Tensor rank of matrix $M = \sum_{i = 1}^k w_i (v_i \otimes v_i)$

I encounter this situation while doing the whitening process of a tensor: If $v_1, \ldots, v_k$ are linearly independent and $w_i \in \mathbb{R}^+$ then the matrix $M = \sum_{i = 1}^k w_i (v_i \...
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Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$ \mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k}, $$ ...
abcd's user avatar
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Implementation of symmetric tensor decomposition algorithm

Context Any symmetric tensor F of rank $d$ and dimension 2 ($F \in S^d\mathbb{C^2}$ for our purpose) can be associated with a homogeneous polynomial $P(F)\in k[x_0,x_1]_d$ in 2 variables of degree $d$....
Baloo's user avatar
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Why does differentiation of a tensor increase its rank?

There is a statement in both Wald and Carroll's GR texts that, in short, state that the derivative of a $(k,l)$-tensor is a $(k,l+1)$-tensor. In both places this as stated as though it should be ...
isaac mandell-seaver's user avatar
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Space of simple tensors

A simple tensor in $V^{\otimes n}$ is one that can be written as $v_1 \otimes \cdots \otimes v_n$ for some choice of $v_i \in V$, these are also called rank 1 tensors. The space of these simple ...
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Confusion about ricci notation

I'm working with an expression of the form: $$x=(A\cdot B)y$$ Where $x$ and $y$ are column vectors of size $(n\times 1)$ and $A$ and $B$ are matrices of size $(n \times n)$ which I know represent ...
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Represent matrix in einstein notation with kronecker deltas

Define an $(n \times m)$ matrix $E_{(i,j)}$ as the matrix which has one element, element $(i,j)$, that is equal to $1$ and the rest equal to $0$. Assume $A$ is an arbitrary matrix of shape $(m \times ...
JDoe2's user avatar
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Rank 1 tensors, how to describe them? (specific case)

I want to undestand a specific case. I consider two $\mathbb{C}$-vectorial spaces, $\mathbb{C}^2$ both. Then, I want to work with $\mathbb{C}^2\otimes \mathbb{C}^2$. Now, I consider basis for each ...
Gyadso's user avatar
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Dimension of the variety of rank 1 decompositions of a matrix

Let $A\in\mathrm{GL}_n(\mathbb{C})$, and let's consider its decompositions into a sum of rank 1 matrices $$A=\sum_{i=1}^t A_i,\ \text{rank}(A_i)=1,$$ where $t\geqslant n$. When $t=n$, Wikipedia claims ...
Sergey Guminov's user avatar
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Rank of Matrix Multiplication tensor $\langle1,n,1\rangle$

$\newcommand{\rank}{\operatorname{rank}}$I am reading through Belzer's Fast Matrix Multiplication, available here. I want to prove tjat $$\rank(t)=\rank(\langle1,n,1\rangle)=n$$ In "usual tensor ...
Louie_the_unsolver's user avatar
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Trying to effect permutating a tensor on its rank

I am reading through Fast Matrix Multiplication by Markus Blaser. I am trying to prove Lemma 5.3 from page 19. It states the following: For any tensor $T\in \mathbb{F}^{n\times m\times t}$, and any ...
Louie_the_unsolver's user avatar
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How to apply tensor symmetry and traceless properties to solve for an unknown quantity in terms of another variable?

To specify, I have the following equation and I am trying to solve for $D_{\ell m}$ in terms of K: $$K_{ijk}=\gamma_{ijk} + (\varepsilon_{ik\ell} \delta_{jm} + \varepsilon_{jk\ell}\delta_{im})D_{lm} + ...
brycon2's user avatar
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How can the isotropic and deviatoric parts of fourth- and fifth-rank tensors be obtained? [closed]

For a second-order rank, we know that the isotropic and deviatoric parts is given by $$ \frac{1}{2} \left( A_{ij} + A_{ji} \right) - \frac{1}{3} A_{ss} \delta_{ij} \, , $$ whereas for a third-rank ...
keynes's user avatar
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Confusion between covariant and contravariant order of a $(n,m)$-tensor.

I read various definitions for a tensor (and watched introduction videos), but still have a small confusion, where I kindly ask for clarification. One definition that I really like and found often is: ...
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Polynomial coefficients tensor generalization

Let $\mathbb{R}_k[x_1, \dots x_n]$ denote the ring of polynomials of degree $k=0, 1 \dots d$. A polynomial is said homogeneous if the non-zero terms all have the same degree. Any homogeneous ...
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Higher order tensors Definition and notation issues

Reading this interesting short paper review https://haggaim.github.io/projects/universality/poster.pdf, I came across some notations/definitions that I'd like to understand better. Let $G \le S_n$ any ...
James Arten's user avatar
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Contraction of tensor?

Given the the tensor $T_{\alpha, \beta...\gamma}=(-1)^n \nabla_\alpha\nabla_\beta...\nabla_\gamma{1\over r_{ab}} $ where $n$ is the rank of the tensor, how does one get to the likes of $T_\alpha={{(r_{...
Stephen 123's user avatar
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293 views

Dyadic (tensor) product of four vectors

I am currently working on a subject, in which the dyadic product of two and four vectors result the second rank and fourth rank tensors as follows: $$ M = a a$$ and $$ \mathbb{M} = a a a a$$ If $...
tio's user avatar
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SVD in scipy and numpy for tensors

Can someone explain to me the difference between SVD of numpy and scipy for Multidimensional arrays (Tensors)? ...
Neuling's user avatar
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Construction of a polynomial with specified integral over several regions

This is appearing in the context of the finite element method. We want to find a $n$-variate polynomial, of order $\leq m-1$ in each variable, such that its integral on each of various subsets of the ...
cohomonoid's user avatar
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are $a_j \otimes a_j$ linearly independent for a frame basis $a_j$?

Consider a full rank matrix $A\in \mathbb{R}^{m\times n}$ with $n\geq m$, let $a_j$ be its $j$-th column then $a_j,j=1,\ldots,n$ span the space $\mathbb{R}^m$, meaning that they are a frame basis of $\...
Chen Ke's user avatar
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How to find the second derivative of an expression in tensor form.

I would like to calculate $\Box\phi$ whereby $\phi = exp(ip_{\mu}x^{\mu})$ and $\Box = \partial_{\mu}\partial^{\mu}$ and whereby $\partial_{\mu} = \frac{\partial}{\partial x^{\mu}} $ and $\partial^{...
DJA's user avatar
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Mathematical properties of Rank-$N$ tensors where $N$>2

WARNING: This question might not have all the necessary tags. I asked about the uses of rank-$N$ tensors in physics on physics stackexchange, but for some reason it was closed saying that my question ...
Rounak Sarkar's user avatar
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Can sum of two (1,1) tensors be written as a (2, 1) tensor?

Tensors can be described as multilinear mappings from r copies of dual vector space (V^*) and s copies of vector space V. Multilinearity means linear in each variable. Thus you can take σ and δ from ...
AloeVera98's user avatar
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2 answers
183 views

Calculation of permutation in tensor of rank 4

I am trying to compute the symmetric part of a 4th order tensor $A_{ijkl}$ From a previous post (Symmetric Part of Product of 2 tank 2 tensors), I saw that I need to compute the permutations of $A_{...
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New to tensors, having problems understanding some basics

I have some components of a contravariant rank two tensor with respect to the standard basis $\{ \boldsymbol{e_1}\ \boldsymbol{e_2} \}$ which are called $a^{ij}$. The task is to find the components $...
SphericalApproximator's user avatar
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239 views

Is the co-skewness matrix a rank-3 tensor?

Variance and skewness are the 2nd and 3rd statistical moments of a random variable's distribution. Unlike the variance-covariance matrix, which is shaped $p\times p$, the skewness-coskewness matrix is ...
develarist's user avatar
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Are rank-3 tensors used as optimization objective functions?

There many optimization problems, like quadratic programming (convex optimization), that use a matrix within the objective function. This is a reference request for any well-known applications where ...
develarist's user avatar
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Are rank 3 tensors always cubes?

a matrix is $A\in \mathbb{R}^{3\times 3}$. It is symmetric and contains 3 row vectors and 3 column vectors containing elements $a_{i,j}$. It looks like a square and, as long as the two dimensions are ...
develarist's user avatar
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How to algebraically express the condition for a tensor to be of rank $1$?

Let $W$ be a finite-dimensional complex vector space and let $V = W\otimes \cdots \otimes W$ ($m$ times). A non-zero element $v \in V$ is said to be of rank $1$ if $v$ can be written as $$v = w_1 \...
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