# Questions tagged [tensor-rank]

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 \$...