Questions tagged [tensor-rank]

For questions about tensor-ranks.

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15 views

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 ...
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1answer
53 views

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 ...
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7 views

What is the transpose of a 3rd rank tensor?

If I have a 3rd rank tensor $\stackrel{\leftrightarrow}{A}$ or in index notation $A_{ijk}$, how to make the transpose of this in index notation? Does the concept of transpose of third rank tensor even ...
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48 views

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 ...
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1answer
49 views

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 ...
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41 views

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} + ...
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1answer
65 views

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 ...
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52 views

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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31 views

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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28 views

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 ...
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1answer
82 views

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_{...
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1answer
105 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 $...
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23 views

rank of a matrix formed from a tensor product

I know that rank of a matrix expressed in term of product of 2 matrices, satisfies the following equation rank(AB) ≤ min(rank(A), rank(B)). When we write a matrix as a product of tensors(of tensor ...
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29 views

CP-ALS algorithm with Kroonenberg and De Leeuw initialization

I am trying to implement the CANDECOMP/PARAFAC - Alternating Least Square algorithm, with a small improvement in the initial guess. The simple initial guess for A is random in the method==0 if block. ...
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1answer
160 views

SVD in scipy and numpy for tensors

Can someone explain to me the difference between SVD of numpy and scipy for Multidimensional arrays (Tensors)? ...
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0answers
24 views

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 ...
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43 views

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 $\...
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1answer
34 views

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^{...
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1answer
109 views

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 ...
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40 views

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 ...
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2answers
92 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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27 views

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 $...
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87 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 ...
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24 views

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 ...
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2answers
821 views

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 ...
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1answer
65 views

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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1answer
42 views

Do orthogonal transformations preserve the symmetry of the tensors?

I have the following doubt: Do orthogonal transformations preserve the symmetry of the 2 rank mixed tensors? It seems logical to me since the symmetry of the tensor needs to be preserved if we change ...
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1answer
516 views

Is the determinant a tensor?

I was reading Schutz's book on General Relativity. In it, he says that a(n) $M \choose N$ tensor is a linear function of $M$ one-forms and $N$ vectors into the real numbers. So does that mean the ...
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1answer
37 views

Tensor contraction and notational problems

I am going through a chapter in my book on tensors, and it gives a basic understanding of tensors. The question posed is simple: "Show that the contracted tensor $T_{ijk}V_k$ is a rank-2 tensor.&...
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2answers
274 views

Dimension of an antisymmetric tensor product space

can somebody explain to me why the dimension of an antisymmetric tensor product space $\Lambda^{r} V$ of rank $r$ and formed from a vector space $V$ with, $\quad dim V = n \quad$ is $\quad {n \choose ...
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1answer
41 views

An array of polynomials can be considered as a tensor?

if $\{1, x, x^2, x^3, \dots, x^{n-1}\}$ is the base of a vector space, can I say that is $\bar{P} = [ P_1, P_2, \dots, P_m]$ a tensor? Where each $P_i$ is a combination of the monomial base. If it's, ...
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0answers
41 views

Approximation in canonical format by rank 1 tensors

Let $I$ be a finite nonempty set and $H_i$ be a pre-$\mathbb R$-Hilbert space and $H:=\bigotimes_{i\in I}H_i$. Let $v\in H$. I would like to show that there is a $u\in H$ with $\operatorname{rank}u=1$...
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1answer
60 views

linear independence of symmetric tensors

I am reading a paper that incidentally uses a bit of theory of symmetric tensor spaces. I came across the following claim: If we're given linearly independent vectors $x_1, \ldots, x_n \in \mathbb{...
3
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1answer
29 views

Diagonalize matrix multiplication

Let $V$ be the space of $2\times 2$ matrices with complex coefficients. Let $A \in V$ and let $L_A:V \to V$, defined by $L_A(X)=A\cdot X$. I am trying to solve the exercise (10) from this book: find a ...
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1answer
64 views

Uniqueness of the matrix representation of tensors

Note that both maps below satisfy the universal property of the tensor product. \begin{align*} \mathbb{R}^2 \times \mathbb{R}^2 &\rightarrow \mathbb{R}^{2 \times 2} &\mathbb{R}^2 \times \...
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1answer
281 views

Finding the irreducible components of a rank 3 tensor

In 3 dimensions, a rank-2 tensor can be identified via a scalar, a vector and a (symmetric, traceless) tensor component by contracting it with $\delta_{ij}$ or $\epsilon_{ijk}$: $$ T_{ij} = \...
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201 views

is a Block matrix a Tensor?

Currently I am starting to study tensor calculus and I came across the definition of the tensor product, and more specifically the definition of tensor rank (ex. a tensor product of 2 rank 1 tensors (...
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2answers
102 views

If a tensor's multilinear rank is $(R,R,R)$, then is its canonical/CP rank also $R$?

Given an order $2$ tensor (i.e. a matrix), one always has that row rank is equal to the column rank, so that its multilinear rank or $n$-ranks is always equal to $(R,R)$ for some $R$. Moreover that $R$...
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1answer
76 views

Dimension of the secant variety $\sigma_2 (\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)$

I am trying to compute the dimension of the secant variety $\sigma_2 (\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)$, for this I want to use the Terracini Lemma, that says that $$ ...
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1answer
138 views

Tensor rank decomposition for some vectors

I have a problem with solving of following problem. We have a vector space $V$ over field $k$ with basis $v_1, v_2$ Rank of a vector $v$ of $V \otimes V \otimes V$ is minimum length of decomposition ...
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1answer
579 views

Intuitive understanding of 2-forms, (1,1)-tensors, and other fundamental objects of exterior algebra or tensor algebra

My background consists mostly of a good level in linear algebra, abstract algebra, undergrad calculus, topology & probability, and some working knowledge of geometric algebra and category theory. ...
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1answer
216 views

Is higher rank tensor always the product of lower rank tensor?

I remember that I saw the definition of tensor somewhere as tensor is an object in $E\otimes F$ for some vector space $E$ and $F$. (here I used an example for rank two.) But most of the time in ...
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2answers
159 views

Tensor confusion - Are coordinates a type of tensor?

(All of this question uses $V$ as the real vector space where tensors are being built from). I am currently trying to learn about tensor notation, and I am running into a road block with ...
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0answers
39 views

Understanding a discrepancy in tensor multiplication

I have seen in texts about quantum computing people take two vectors and do tensor multiplication. Now, what confuses me is that vectors are (1,0)-tensors. This means that when I multiply two of them, ...
3
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1answer
108 views

Does the kernel of every alternating form contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $1 < k < n$. Let $\alpha \in \bigwedge^k (V^*) \cong (\bigwedge^k V)^*$. Thinking of $\alpha$ as a linear functional $\bigwedge^k V \to \...
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1answer
187 views

Tensor calculation: outer product

it is given a tensor: $T=\begin{pmatrix} 1\\ 1 \end{pmatrix}\circ \begin{pmatrix} 1\\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\\ 1 \end{pmatrix}+\begin{pmatrix} -1\\ 1 \end{pmatrix}\circ\begin{pmatrix} ...
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1answer
276 views

Why a vector is a (1,0) tensor?

I am looking for some familiar examples of Tensors and I am wondering why a vector is a (1,0) tensor type? That is it takes some covector and gives and scalar!! How?
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2answers
142 views

Abelian group Rank ιs equal to the dimension of the tensor product [closed]

If $G$ is a finitely generated abelian group then why it's rank is ιs equal to the dimension of $G\otimes_\mathbb{Z} \mathbb{Q}$ as a vector space over $\mathbb{Q}$? Thank you in advance.
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1answer
4k views

How many dimensions will a derivative of a 3-D tensor by a 4-D tensor have? [closed]

As the title above, I find it hard to imagine or illustrate. It is a question from Coursera.
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1answer
33 views

Scalars in tensor notation

I know that usually the number of indices on a tensor indicates it’s rank, so how do you represent a scalar/rank zero tensor. I’ve had trouble making this work and it seems at times like there’s ...