# Questions tagged [tensor-products]

For questions about tensor products, which allow us to build "linear" objects from "multilinear" ones. Add other specific tags to indicate the subject you're referring to.

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### Counterexample for short exact sequence of tensor product of modules

Let $R$ be a semisimple Artinian ring and $A, B, C$ be right $R$-modules such that $A \subseteq B$ and $B/A \cong C$. I have managed to show that $(B\otimes M)/(A\otimes M) \cong C\otimes M$ for any ...
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### Meaning of wedge product to calculate volume, and basic concept of tensor [closed]

I heard that wedge product is correct type to calculate volume form from professor and here https://mathworld.wolfram.com/WedgeProduct.html And he said that to calculate surface integral int(FdS)= int(...
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### How exactly do tensor products correspond with “tensors” as understood in tensor calculus & mathematical physics?

This is a mathematical construct that I've found rather hard to construct a more intuitive description of than what is usually given - heck, I've sooner with enough digging found what I'd consider a ...
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### 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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### Tensor Product of Matrix and Vector

I am studying tensor networks and tensors. A commonly described operation is the tensor product (denoted by $\otimes$) which is a generalization of the outer product (as I understand it). It makes ...
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### $\mathbb{R}^{\Omega}\otimes\mathbb{R}^{\Omega}\cong\mathbb{R}^{\Omega\times\Omega}$ and beyond

Notations : $V^*$ is the dual of $V$, $\otimes$ is tensor product of vector spaces/modules, $\mathbb{R}^{\Omega}$ is the set of functions from $\Omega\subset\mathbb{R}$ to $\mathbb{R}$. I found in Lee'...
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### If $V$ is simple then is $V\otimes W$ is simple given $W$ is simple?

Let $W$ be a simple, finite dimensional, $\mathbb{C}G$-module and $G$ is a finite group. Suppose $V$ is also a simple, finite dimensional, $\mathbb{C}G$-module, would it be the case that $V\otimes W$ ...
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### Base of tensor product

Given the tensor product $V_1 \otimes V_2$ (so we have a bilinear map $\otimes : V_1 \times V_2 \to V_1 \otimes V_2$) and bases $B_i$ of $V_i$ (not need to be finite) how to prove using only the ...
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### Diagonalisation of tensor product

Say I have two diagonalisable matrices $A$ and $B$, diagonalisable in different basis'. \begin{equation} A=\sum_ia_i|a_i\rangle\langle a_i|\qquad\text{and}\qquad B=\sum_ib_i|b_i\rangle\langle b_i|\;. \...
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I’m used to the abstract definition of tensor product as an universal solution, but I’m getting trouble in how to calculate one: Take an example, I think $m\mathbb{Z}\otimes \mathbb{Z}/(m)=0$, since $... 1answer 48 views ### Why doesn't right exactness of a functor (say tensor product) imply exact? Let's work in$R$-mod for the remainder of this question. I know I'm probably missing something super basic, but here goes. Let$F$be the functor$- \otimes_R M$for some$R$-module$M$. Let's use ... 2answers 34 views ### Prove that there is an$R$-module isomorphism between$Q \otimes_R N \cong N$, Q is a quotient field of$R$. Let$R$be an integral domain with quotient field Q and$N$be a unitary, divisible, torsion-free left$R$-module. Show that there is an$R$-module isomorphism so that$Q \otimes_R N \cong N$. Here ... 1answer 22 views ### How to make a multidimensional SVD? Is it possible to define a tensor Singular Value Decomposition (SVD)? For example for the 3 tensor $$\left[\begin{array}{rr}1&-1\\1&1\\\hline 1&-1\\-1&-1\end{array}\right]$$ Can be ... 1answer 37 views ### unique factorization of product of linear functionals Let$V$be a vector space over the complex numbers. Let$f_{1\leq i \leq n}$and$g_{1\leq i \leq n}$be two sets of nonzero linear functionals on$V$. Suppose we have $$f_1(v) f_2 (v)\ldots f_n (v) = ... 1answer 71 views ### Is that a exact sequence? Let A be an Algebra, S a subalgebra ans W a subspace s.t SW \subset W, WS \subset W and S,W generate A. Show that for any A-module M the follow sequence is exact.$$A \otimes_S W \... 2answers 35 views ### Unique definition of$T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$We know that$V \otimes(W \otimes U) \cong (V \otimes W) \otimes U \cong V \otimes W \otimes U$but as far as I know there is not a canonical isomorphism. Now consider$T(V) = \bigoplus_{n \in \...
Let $A: V\to V$ and $B: W\to W$ be linear operators on vector spaces $V$ and $W$. I know how to prove $$\operatorname{tr}(A\otimes B) = \operatorname{tr}(A)\operatorname{tr}(B)$$ by appealing to a ...
We know that if $G$ and $H$ are finite groups and $F_p$ is a field of characteristics $p$, then $$F_pG\otimes_{\mathbb{F}_p} F_pH\cong F_p(G\times H).$$ Here $\otimes_{\mathbb{F}_p}$ denotes the ...