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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.

2
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1answer
19 views

Isomorphism between tensor product of modules and quotient module

I'm trying to show that $M\otimes_{R}R/\mathbf{m}$ and $M/\mathbf{m}M$ are isomorphic as $R$-modules, where $M$ is an arbitrary $R$-Module, $R=k[x_{1},\ldots,x_{n}]$ with a field $k$, $\mathbf{m}=(x_{...
1
vote
1answer
22 views

Algebraic tensor product as a quotient space

I am trying to get a better understanding of algebraic tensor products. The following definition comes from page 47 of Weidmann's Linear Operators in Hilbert Spaces. Let $H_1$ and $H_2$ be vector ...
0
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0answers
10 views

Topology on space of full signatures in rough path theory

In Theorem 3.1 of this paper, the following result is formulated: Suppose $f:S_1 \to \mathbb{R}$ is a continuous function where $S_1$ is a compact subset of $S(\mathcal{V}^p(J,E))$. Then for any $\...
0
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0answers
19 views

the matrix of the billinear form: $ T = e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$

Let $B = ((1,2)^T,(1,3)^T)$ be the basis of $V=\Bbb R^2$. Find the dual basis $B^*=(e_1,e_2)$ Find the matrix of the billinear form: $ T = e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$ ...
0
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1answer
12 views

Most elegant way to show Inner-Product of tensors with indices?

How would you write this sum most elegantly? (Or in other words: What is the most elegant and best way to show a sum of products with indicies?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
0
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0answers
20 views

Are Hypermatrices equivalent to Tensor?

One number $a$ can be seen as a one-dimensional matrix. Can we generalize matrices in a high-dimension sense? Think of a “cubic matrix”, which looks like a crystal with a number attached to its ...
0
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1answer
24 views

Let $T \in T_{2}^{1}$ be associated with the map given as $Z(A,B) = AB - BA$, where $A,B \in V$, find $[T]_B$.

Let $V$ be the space of all $3 \times 3$ antisymmetric matrices. And let $T \in T_{2}^{1}$ (tensors) be associated with the map given as $Z(A,B) = AB - BA$, where $A,B \in V$. The basis of $V$ is ...
0
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1answer
42 views

Why is $\mathbb{Z}\otimes Ag = Ag$, where Ag is an abelian group

Why is $$\mathbb{Z}\otimes Ag = Ag,$$ where Ag is an abelian group? In other words, why is it an identity? Perhaps an explanation of the simpler example of the $\mathbb{Z}\otimes \mathbb{Z}$ is equal ...
4
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0answers
52 views

Kronecker Product Interpretation

The algebraic expression for a Kronecker product is simple enough. Is there some way to understand what this product is? The expression for matrix-vector multiplication is easy enough to understand. ...
1
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0answers
33 views

Special case of isomorphism $S\otimes_R \mathrm{Hom}_R(M,N) \simeq \mathrm{Hom}_S(S\otimes_R M, S\otimes_R N)$

$R$ is commutative ring, $S$ is commutative $R$-algebra and $M$ is $R$-module. So we have the S-module isomorphism $$S\otimes_R \mathrm{Hom}_R(M,N) \simeq \mathrm{Hom}_S(S\otimes_R M, S\otimes_R N)$$ ...
2
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1answer
43 views

Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
0
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0answers
14 views

Tensor product of modules finitely generated

Let $S$ be extension of $R$ and $M$ is an $R$-module. If $M$ is finitely generated of $R$-module then $S\otimes_RM$ is finitely generated of $S$-module. Proof We can construct a ring homomorphism $f:...
2
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1answer
64 views

Is the Evaluation map of an R-Module of rank 1 and hom injective

This is in context of a larger problem of showing that the dual of an invertible sheaf is invertible on a scheme. I want to show that given a free R-module A of rank 1, the standard evaluation map ...
0
votes
1answer
11 views

summation notation for the tensor product of two operators

Suppose there exists a linear operator $A$ from the vector space $V$ to $V'$ and a linear operator $B$ from the vector space $W$ to $W'$. I.e., $A:V \rightarrow V'$ and $B:W \rightarrow W'$. The ...
1
vote
1answer
48 views

Global dimension of tensor product of algebras

I am looking for a reference or proof of the following fact: If $K$ is algebraically closed field and $A$ and $B$ are finite dimensional $K$-algebras then $\text{gl.dim}(A\otimes_KB)= \text{gl.dim}(A)...
0
votes
1answer
36 views

The exact sequence of tensor product

Prove that for all free right $R$-module $F$ and for all exact sequences of $R$-modules $$0\to M\xrightarrow{f}N\xrightarrow{g}P\to 0$$ then $$0\to F\otimes_RM\xrightarrow{1\otimes f}F\otimes_RN\...
0
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0answers
7 views

Product of dyadic and vector

In wikipedia for product of dyadic and vector equality, it is given that: $\overrightarrow {c}.(\overrightarrow {a} \otimes \overrightarrow {b})=(\overrightarrow {c}.\overrightarrow {a}) \otimes \...
0
votes
1answer
11 views

Distributivity of tensor product over a direct sum

Let $\mathcal{H}, \mathcal{K}$ be finite dimensional Hilbert spaces and consider the space $$\left(\mathcal{H} \oplus \mathcal{K}\right) \otimes L^2(\mathbb{R}).$$ I would like a reference to show ...
2
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0answers
35 views

Epsilon tensor product of locally convex spaces

I want to understand the definition of the $\varepsilon$-tensor product of two locally convex vector spaces. (Mainly as a hobby.) Let $X,Y$ be locally convex vector spaces and let $B(X^{\ast},Y^{\...
4
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2answers
427 views

Is the tensor product (of vector spaces) commutative?

I've just learned a bit about the tensor product and I couldn't find a real answer to this. I've read something about, that in some cases it could be or not. Let's consider next example: In the ...
1
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0answers
27 views

Why is $ \mathbb{K}[X] \otimes_{\mathbb{K}} \mathbb{K}[Y] \longrightarrow \mathbb{K}[X \times Y] $ surjective?

I am studying products in the category of affine varieties and I don't know how to prove that the map $ \mathbb{K}[X] \otimes_{\mathbb{K}} \mathbb{K}[Y] \longrightarrow \mathbb{K}[X \times Y], $ ...
3
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0answers
79 views

Invariants restricting possible similarity matrices to only permutations for graph isomorphism problem?

Testing $Tr(A^k) = Tr(B^k)$ for $k=1..n$ and $n\times n$ matrices ensures their similarity - existence of orthogonal matrix $OO^T =I$, such that $A=OBO^T$. In graph isomorphism problem we ask if such ...
0
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2answers
24 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.
2
votes
1answer
48 views

Isomorphism of Ideal tensored with affine open and restriction of ideals

Let $f:X \rightarrow Y = \operatorname{Spec}A$ be a morphism and $Y = \bigcup U_\alpha$ where $U_\alpha = \operatorname{Spec}A_\alpha$. Given the ideal $I = \{a\in A: f^*(a) = 0\}$, show that $I \...
2
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0answers
35 views

Inner product of $k$-forms

I'm working on the following problem from Lee's Introduction to Riemannian Manifolds: Let $(M,g)$ be a Riemannian $n$-manifold. show that for each $k=1,\ldots, n$, there is a unique fiber ...
0
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0answers
20 views

orthogonal complement of tensor product

I am studying Functional analysis. And I don't understand an equation that is followed as below : If $S_1$ and $S_2$ are vector spaces, then $(S_1 \otimes S_2)^\perp = (S_1 ^\perp \otimes S_2) + (...
0
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0answers
17 views

Tensorproducts of modules

I have some exercises on tensor products of modules and algebras and I'm a little bit confused. This is our definiton of a tensor product: If $X$ is a right $R$-module and $Y$ is a left $R$-module, ...
1
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0answers
18 views

Coordinate Derivative of a Tensor Product

I know similar questions have been asked on the stack exchange, but all the ones I've found haven't had answers. I have just started working with tensor products, but have had to learn them a little ...
9
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0answers
46 views

The “semi-symmetric” algebra of a vector space

If $V$ is a vector space over a field $K$ then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\in ...
3
votes
3answers
65 views

Tensor is non-zero

Let $A,B$ be $k$-algebras, where $k$ is a field. Let $a_1,a_2,\dots a_n$ be elements of $A$, which are linearly independent over $k$. Similarly let $b_1,b_2, \dots, b_n$ be elements of $B$, which are ...
0
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0answers
25 views

Find example to show that the containment $\text{Supp}(M \otimes_R N) \subset \text{Supp}\, M \cap\text{Supp}\, N$ is proper [duplicate]

$$(M \otimes_R N)_P\cong(R-P)^{-1}(M \otimes_R N)\cong(R-P)^{-1}M \otimes_R (R-P)^{-1}N\cong M_P \otimes_R N_P$$ This implies $\text{Supp}(M \otimes_R N) \subset\text{Supp} M\, \cap \text{Supp}\,N$. ...
1
vote
1answer
57 views

Differentiation of tensor product

I have a tensor equation $$\frac{\partial A_{ij}}{\partial B_{kl}}=\frac{\partial A_{ij}}{\partial C_{pq}}\frac{\partial C_{pq}}{\partial B_{kl}} $$ $C_{pq}$ can be written as $C_{pq}=B_{pq}+aB_{mm}\...
0
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0answers
30 views

Contraction of two symmetric and an antisymmetric tensor

Given that $a_{ij}$ and $b_{pq}$ are two symmetric tensors, and $c_{yz}$ is an antisymmetric tensor, is the following true? $a_{ij} c_{ik} b_{kj} = 0 $ If yes, how?
0
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0answers
21 views

Decomposition of permutations and wedge products.

Let $V$ be an $\mathbb{R}$-vector space. Denote the space of all alternating $k$-linear forms from $V^k$ to $\mathbb{R}$ by ${\cal A}_k(V, \mathbb{R})$ Suppose $f\in{\cal A}_p(V, \mathbb{R})$ and $g\...
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0answers
9 views

Projective tensor product on projective LCM is exact

I am reading the book "The Homology of Banach and Topological Algebras" by A.Y. Helemskii and couldn't understand one lemma on page 204 about complex splitting. I understood how to prove that complex ...
1
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1answer
43 views

Unique $\mathbb{R}$-linear map between tensor products

I am reviewing materials in tensor products and I got stuck on this one, and I am never comfortable with "showing there exists a unique linear map" type of question. Let $V$ be a real vector space ...
1
vote
1answer
27 views

Tensor products of rings as modules over themselves

Let $R$ be a non-commutative ring, and, consider the tensor product $$R \otimes_R R $$ where we consider the 'first $R$' as a right module over $R$ and the 'second $R$' as a left module over itself. ...
0
votes
2answers
33 views

Properties of outer product of two unit vectors? Why is there only one non-zero eigenvalue for such a matrix?

Let $x,y$ be two unit vectors. $A=xy^T$ be the outer product. The eigenvalues of A are seen to be $[0, 0, 0,...0, k]$. Why is that? What are the properties of the outer product of two unit vectors? ...
1
vote
1answer
62 views

Push-forward of a tensor

Let $T^\mu_\nu \frac{\partial}{\partial x^\mu} \otimes \mathrm{d}x^\nu$ be a tensor field of type $(1,1)$ on $M$ and let $F:M\to N$ be an diffeomorphism. Show that the induced tensor on $N$ is $$F\...
3
votes
1answer
77 views

The set of bilinear forms is a right $(R \otimes R)$-module

Let $V$ and $A$ be abelian groups. An $A$-valued bilinear form on $V$ is a $\mathbb{Z}$-module homomorphism $$\beta : V \otimes_{\mathbb{Z}} V \rightarrow A$$ Now, let $V$ be a left $R$-module, where $...
0
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1answer
13 views

Example of a bilinear form of abelian groups

Let $X$ and $Y$ be abelian groups. Then, a $Y$-valued bilinear form on $X$ is a $\mathbb{Z}$-module homomorphism $$\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$$ How does this relate to the standard ...
3
votes
1answer
46 views

Tensor products of modules over non-commutative rings

I've been learning about tensor products over modules, but where the ring acting on the module is commutative. When $R$ is non-commutative, we consider a right $R$-module $M$ and a left $R$-module $N$...
1
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1answer
17 views

Tensor product of a bimodule (projective as a right module) and a projective right module is projective

Let R and S be rings. Let L be a projective R right module ($L_R$) and M be a R-S bimodule ($_RM_S$) such that M is projective as a S right module. I'm trying to prove that $$(L \otimes M)_S$$ is ...
2
votes
1answer
98 views

Canonical ring structure on the tensor product $R \otimes_\mathbf{Z} S$.

Let $R$ and $S$ be commutative rings. I need to show that there is a unique ring structure on the $\mathbf{Z}$-module $T := R \otimes_\mathbf{Z} S$ such that $$ (r_1 \otimes s_1)(r_2 \otimes s_2) = ...
1
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0answers
17 views

The action of tensor product over N terms on a ket.

Equation (6) of the paper titled, Multi-player and Multi-choice quantum game has left me puzzled-after many hours-as to how it is being derived. My working begins from the generic form seen just after ...
1
vote
1answer
31 views

Tensor product of module homomorphisms is element of tensor product or mapping?

Here: $A$ is a ring. $E,E'$ are right $A$-modules, $F,F'$ are left $A$-modules. $u:E\rightarrow E'$ and $v:F\rightarrow F'$ are $A$-module homomorphisms. With this setup, I didn't understand ...
2
votes
1answer
72 views

Example of tensor product of two representations

This is similar to what Serre wrote in his book on linear representations of finite groups by Serre: There is also the tensor product which has the properties of "multiplication". Let $V_1$ and $...
0
votes
2answers
41 views

Equivalence between these tensor product definitions

Let $V$ and $W$ be vector spaces. Then the tensor product $V \otimes W$ of $V$ and $W$ is the vector space $V \otimes W$ together with a bilinear map $\phi: V \times W \rightarrow V \otimes W$ such ...
0
votes
1answer
28 views

When do tensor products of elements coincide

Let $M,N$ be $R$-modules and $m \otimes n, m' \otimes n' \in M \otimes_R N$ non-zero (EDIT) elements. When does $m \otimes n = m' \otimes n'$ hold? Obviously, this is true if either $(m',n)=(rm,rn')$ ...
1
vote
2answers
29 views

Tensor product notation as a power

I chance upon a notation while reading a paper which I do not quite understand. Suppose that $\hat{J}$ is a operator in a tensor product of two N$ dimensional Hilbert space. Explicitly, it is given ...