Questions tagged [tensor-rank]

For questions about tensor-ranks.

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An application of tensor products [closed]

To prove is the following sentence:A rectangle with the edge lengths $a $ and $b$ is divided into $n $ smaller rectangles with the Edge lengths $a_i$ and $b_i$ decomposed.For each $ i = 1. . . n$ ...
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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
37 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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38 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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22 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{...
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1answer
19 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
55 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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22 views

Finding the irreducible components of a rank 3 tensor

In 3 dimensions, a rank-3 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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48 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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14 views

Operations in tensor algebra: unfolding and contraction

For my study the tensor algebra I am using the following two papers: 1 [Era of Big Data Processing by Andrzej CICHOCKI]1 2 [Tensor Decompositions and Applications by T KOlda]2 My questions: Tensor ...
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2answers
50 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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53 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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22 views

Orbit closures of real symmetric bilinear forms

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^n)$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$. I ...
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1answer
53 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
139 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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19 views

Proving the existence of a tensor with rank at least $n^2/3$

In this post on MO, the asker says from dimension argument it easily follows that there exists a tensor of tensor rank at least $\frac{1}{3}n^2$. I'm not sure what "dimension argument" the asker ...
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1answer
73 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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1answer
55 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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35 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, ...
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1answer
50 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
95 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
120 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
62 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
2k 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
24 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 ...
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35 views

Tensor cross product proof: $Sa \times Sb = \det(S)S^{-T}(a \times b)$ [duplicate]

I need to prove $$Sa \times Sb = \det(S)S^{-T}(a \times b),$$ given that $a$ and $b$ are vectors and $S$ is a second-order (rank $2$) tensor. I have the hint: vectors $u=v$ iff $u\cdot a=v\cdot ...
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1answer
511 views

Upper bound CP tensor rank

I have a question about CP tensor ranks. In the following, $\mathcal X \in \mathbb R^{n_1 \times n_2 \times n_3}$ is a third-order tensor of CP rank $R$, i.e., there exist vectors $a_i$, $b_i$ and $...
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2k views

What is a rank-1 tensor? What is the meaning of rank in this context?

I feel like different sources use the term "rank" differently, which is perhaps leading to my confusion. When I think of rank I think of number of linearly independent columns/rows, number of pivots ...
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1answer
69 views

Finding non-singular transformation mapping one tensor to other in $(\Bbb F_2)^{\otimes 3}$

Let $u, v \in V\doteq \mathbb{F}_2^{2 \times 2 \times 2}= \mathbb{F}_2 \otimes \mathbb{F}_2 \otimes \mathbb{F}_2$ be given by $$u = e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_1 + e_1 \...
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1answer
74 views

How do I tell the rank of the electric susceptibility tensor (and others)?

I understand that a tensor is a multilinear map from $V^*\times\cdots\times V^*\times V\times\cdots\times V$ to $V$'s underlying field, where $V$ is a vector space and $V^*$ its dual. This is fine, ...
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2answers
822 views

Tensor Dot Product of two tensors of arbitrary order

I am currently working on implementing the inner(scalar or dot) product of two tensors of arbitrary order. As far as i understand, you need to make sure, that the last dimension of the first tensor $\...
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1answer
59 views

Convert from one tensor canonical form to another

Suppose we have two canonical forms $A, B \in \mathbb{F}_2^{2 \times 2 \times 2}$ of a 3-dimensional tensor product space over the finite field with two elements, where $A = e_1 \otimes e_2 \otimes ...
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2answers
119 views

tensor rank of an element in a tensor product

Let $V$ and $W$ be finite-dimensional vector spaces over $k$ with $\text{dim}(V)=n$ and $\text{dim}(W)=m$. How can I see that every element $t \in V \otimes_k W$ has tensor rank at most $\text{min}\{...
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0answers
149 views

Show that the derivative of a second order tensor gives a third order tensor

Let $U_{i,j}$ be a second order tensor. Show that $\frac{\partial U_{i,j}}{\partial x_{k}}$ is a third order tensor. I know how to prove that the gradient of a scalar field (which is a tensor of ...
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1answer
73 views

Maximal rank of tensors in $F^n\otimes …\otimes F^n$.

What is the largest possible rank of a tensor in the space $F^n\otimes ...\otimes F^n$ where we have $k$ copies of $F^n$? It is quite easy to see that it is at most $n^{k-1}$ (I have commented the ...
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120 views

Maximum tensor rank

According to wikipedia, it is unknown what the maximal rank of a tensor is in the space $F^{n_{1}} \otimes ... \otimes F^{n_{k}}$. It is quite easy to show that it is at most $$\prod _{i=1} ^{k} n_{i}...
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Prove upper bound inequality for the dimension of the space of symmetric tensors

I want to check that the dimension of the space of symmetric tensors $N(n,m) := dim(Sym^m(\mathbb{R}^n))$ satisfies $N(n,m) \leq \frac{n^m}{m!}(1+\frac{2m^2}{n})$. Thus I need prove inequality. If $...
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230 views

What is a tensor?

I'm trying to understand a paper that works with tensors. I understand that a tensor of rank 0 is a scalar, a tensor of rank 1 is a vector, and a tensor of rank 2 is a dyad (and therefore it can be ...
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3answers
95 views

Can a Rank Three Tensor act as a Trilinear, Bilinear, or Linear Map?

Can a rank three tensor act as a trilinear, bilinear, and linear map? Similarly, a matrix (a representation of a rank two tensor) can be bilinear, taking in two vectors and spitting out a scalar for ...
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4answers
488 views

How is a (0,0) rank tensor a number?

I've always been confused by this. If tensors are basically functions whose inputs are vectors/convectors (tensors) and outputs are numbers (or other nth-ranked tensors), then how does a tensor that ...
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1answer
74 views

Weird Notation for Trace of an Endomorphism

I am having some difficulty understanding a piece of notation from Riemannian Geometry: and Introduction to Curvature by John M. Lee. In Section 2 just under equation 2.3 Lee defines the trace ...
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1answer
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Could a Rank Two Tensor be a Scalar?

The universal property of the tensor product (of vectors or vector spaces) states that a bilinear map out of a cartesian product is a linear map out of the tensor product. I've also heard that tensors ...
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1answer
3k views

Actual example of tensor contraction

So I'm having trouble to compute tensor contractions with "actual" numbers from the matrix representations of the tensors. I have only seen abstract theoretical examples on the internet so I'm asking ...
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1answer
271 views

Uniqueness of Tensor Decompositions (Aren't Matrix Decompositions a Special Case?)

It seems that higher-order tensors (of order 3 or higher) generally have unique decompositions under relatively mild conditions. For example, Kruskal proved that if an order-3 Tensor $T$ can be ...
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1answer
116 views

tensor product, isotropic

Assume that $T_{ijkl}$ is a 4-th rank isotropic tensor. Why is it $\epsilon_{ijk}T_{ijkl}=0$, without assuming the general formula for a 4-th rank isotropic tensor $\alpha \delta_{ij}\delta_{kl}+\beta\...
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99 views

How to find eigenvalues and eigenvectors of higher-order tensors?

I'm currently looking a way to find the eigenvalues and eigenvectors of a tensor which has a higher order than 2.
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1answer
59 views

For all rank two tensors, is $A:BC = AB^T:C$?

Suppose that $A$, $B$, $C$ are rank two tensors that are not necessarily symmetric, and I have a contraction as below. Is the following equivalent? $A:BC \equiv AB^T:C$ If not, what is the correct ...
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217 views

Why are the covariant and contravariant components of the metric tensor defined this way?

The metric tensor, using covariant components is $g_{ik}$ = $e^{(j)} \cdot e^{(k)}$ and using contravariant components it's $g^{ik}$ = $e_{(j)} \cdot e_{(k)}$. This seems counterintuitive to me. Why ...
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Manipulating tensor products

I'm trying to evaluate $\mathbf{M}(\vec{e}_x)$, where $$\mathbf{M}=\tilde{\omega}^x\otimes\vec{e}_x +\tilde{\omega}^x\otimes\vec{e}_y+\tilde{\omega}^y\otimes\vec{e}_y.\tag{1}$$ Here's my ...
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26 views

3+ Dimensional Matrices

I want to describe all polynomials that have the basis $x_1^{d_1}x_2^{d_2}...x_n^{d_n}$ where $\sum_i d_i = k$, $d_i\geq 0$, $d_i\in \mathbb{Z}$. I want to encode this in a nice way. For example, if $...