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Questions tagged [tensor-rank]

For questions about tensor-ranks.

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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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Tensors that don't change in different reference frames

So for example for tensor of 1 rank its 0 vector. We can show it by just rotating system and if components are equal in both frames it means they equal 0.For tensor of 2 rank we write it as sum of ...
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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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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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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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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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How to solve the tensor approximation optimization problem?

Given a third-order tensor $\mathcal{X}\in\mathbb{R}^{I_1\times I_2\times I_3}$, we want to find an approximation tensor $\hat{\mathcal{X}}$ of $\mathcal{X}$ with $R$ rank-one components, and some ...
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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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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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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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How to derive a bound of distortion / error between two different tensor decompositions.

Consider a tensor $\mathcal{X}\in\mathbb{R}^{I\times J\times K} $. It can be approximately decomposed/factored in multiple ways. Namely by using the TUCKER3 decomposition: $\mathcal{X}\approx \sum_{p=...
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differential of a contracted product with a 3 order tensor and uniform vectors

I would like to know if expression of $\text{d}(u^{r}_{st}\,a_{r}\,b^{s}\,c^{t})$ is equal to zero. Indeed, I consider a 3 order tensor (actually (1,2) tensor) $u^{r}_{st}$ contracted with uniform ...
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49 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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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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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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Bounds on tensor rank

What are the results on maximal and expected rank of tensors in $F^{n}\otimes ...\otimes F^{n}$ $k$-times?
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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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Divergence of a third rank tensor with non symmetric connection

I need to construct a 2 rank tensor with covariant derivative of a 3 rank tensor. Like that: $${F^{\mu}}_{\nu} = \frac{1}{2} \nabla_{\rho} {{J^{\mu}}_{\nu}}^{\rho}$$ Thus $${F^{\mu}}_{\nu} = \frac{...
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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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Can any tensor with real values be decomposed exactly with a Canonical Polyadic Decomposition?

Let $T \in \mathbb{R}^{I_1 \times ... \times I_d}$. Is it true that $T$ can always be written with a sum of rank-one tensors: $T = \sum_{i = 1}^r a_1 \circ ... \circ a_d$ for some real-valued vectors ...
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Matrix concatenation and its symbols

Somewhat related to this post(concatenating vectors to form a matrix). Create a tensor with multiple matrices. How do you call this operation? For example, $$\alpha_{abc}=\begin{cases} A_{ab}, &...
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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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Rank decomposition linear independence

Let $f:U\rightarrow V$ be a linear map, or alternatively $f\in U^{\ast}\otimes V$. It is clear that if $\textrm {rk}\left(f\right)=r$ and $$f=\sum_{k=1}^{r}\phi_k\otimes v_k$$ with $\phi_k\in U^{\ast}...
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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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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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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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702 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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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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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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Are closest low rank tensor approximations of nonnegative tensors nonnegative?

Let $W_{i_1,i_2,\ldots,i_d}$ be an order $d$ tensor consisting of real valued entries. I want to consider the approximation of $W$ by some other tensor $\hat{W}$ in frobenious norm $||W-\hat{W}||^2=\...
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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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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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3rd rank/mode tensor unfold operation in Python

I am trying to use the reshape command in numpy python to perform the unfold operation on a 3rd-mode tensor. I'm not sure whether what I'm doing is correct. I found this paper online Tensor ...
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Proof that del operator is a rank 1 tensor

In my maths course we have gone over the following proof that $\frac{\partial}{\partial x_i }$ transforms as a rank 1 tensor: Let $$ x_i' = L_{ij} x_j$$ it then follows that $$\frac{\partial x_i'}{\...
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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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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 $...
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Tensor rank as a first order formula

I heard in a talk today that the problem of tensor rank over $\mathbb{Q}$ is not even known to be decidable and it is equivalent to the existential theory of over that field. Did not interrupt the ...
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If $A_{ij}$ is rank $2$ tensor, show that $\frac{\partial(A_{ij})}{\partial x_k}$ is a rank $3$ tensor.

If $A_{ij}$ is grade $2$ tensor, show that : $$\frac{\partial(A_{ij})}{\partial x_k}$$ is a grade 3 tensor. Solution : $$\frac{\partial(A_{ij})}{\partial x_k}= \frac{\partial(A_{ij})}{\...
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779 views

Anti-symmetric tensor of second order from a vector

When given a vector $\overrightarrow V$ = $(x, x+y, x+y+z)$. Find the second order antisymmetric tensor associated with it. This problem needs to be solved in cartesian coordinate system. The problem ...
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Tensor product and linear dependence of vectors

Let $V_1, \ldots, V_k$ be complex vector spaces. Given $k$ vectors $v_1 \in V_1, \ldots, v_k \in V_k$, we define that the tensor product $v_1 \otimes \ldots \otimes v_k$ has rank 1. For any tensor $T \...
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Eigenvalues of a contracted 4-tensor

Let $T^{ij}_{i'j'}$ be a 4-tensor, $1 \leq i,i' \leq n$, $1 \leq j,j' \leq m$, $n$ and $m$ may differ. That is, the first (second) covariant coordinate is of the same dimension as the first (second) ...
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What is the third aspect (component) of a 2nd-Order Tensor?

I am just starting to become familiar with Tensors (most familiar with Moment of Inertia Tensor & Spring Constant Tensor), I am trying to understand the fundamental nature of them (as High-Level ...
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Anti-Symmetic Tensor Operation rank 4

I have a problem opening up an antisymmetric tensor. Please pardon my jargon I may not express it the best way. I need to open up: $T^{bk}_{\enspace \enspace [q}T^{df}_{\enspace \enspace r]}$ This ...
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Definition of independent components of a tensor

How can the number of independent components of a tensor $T_{i_1 i_2 \dots i_p}$ defined? Example: Let us consider a symmetric matrix $A$ such that $A{}_{ij} = A{}_{ji}$, and hence in $n$ dimensions ...
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Problem regarding tensor

Prove that the contraction of a tensor $A^l_m$ is a scalar or invariant. My try: Mixed tensor of rank 2 formula: $$\bar A^l_m=\frac{\partial \bar x^l}{\partial x^p}\frac{\partial x^q}{\partial \bar ...
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A problem regarding tensors

If $T_i$ are the components of a covariant vector, show that $\left(\frac{\partial T_i}{\partial x^j}-\frac{\partial T_j}{\partial x^i}\right)$ are the components of a skew-symmetric tensor of rank 2. ...
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k-dimension of tensor product of two modules over a k-algebra

This is a generalization of the currently unanswered question here. Let $k$ be a field, $A$ be a finite-dimensional $k$-algebra, and $M$, $N$ right and left $A$-modules, respectively, both finite ...
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Write down an explicit tensor of border rank $r$ in $\mathbb{C}^r \otimes \mathbb{C}^r \otimes \mathbb{C}^r$ with rank greater than $r$

This question is from the text Geometry and Complexity Theory, from J.M. Landsberg. Before start talking about the question, I think it is good to show the definition of border rank. Consider a tensor ...