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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12 views

An isomorphism defined on the tensor product of two quotient modules

Let $A$ be a ring, $E$ be a right $A$-module and $F$ a left $A$-module. Let $E'$ be submodule of $E$ and $F'$ a submodule of $F$ with the canonical injections $i:E'\rightarrow E$ and $j:F'\rightarrow ...
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32 views

Can we recover the bases of two infinite-dimensional vector spaces into a tensor product?

I know that in general, if V,W are infinite-dimensional vector spaces with basis {${v_i}$} and {${w_j}$} respectively it holds that V⊗W has as basis {${v_i⊗w_j}$}. My question is: what about the ...
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18 views

Isomorphism between tensor product of Hilbert spaces

While I am familiar with the isomorphism between finite-dimensional vector spaces with the same dimension and isometric isomorphism between Hilbert spaces, I realize I am getting confused by $n$-th ...
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Tensor products and direct integrals of von Neumann algebras

Please tell me a reference that includes a theorem like: For a measure space $(\Omega,\mathcal{B},\mu)$ and von Neumann algebras $N,M_\omega,\, \omega\in\Omega$, $$ N\overline{\otimes} \int^{\oplus}_\...
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1answer
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Not every matrix on V ⊗ W can be written as a tensor product of a matrix on V and another on W.

I am reading some notes about tensor product of vector spaces (those in here) where the following sentence can be found: Note that not every matrix on V ⊗ W can be written as a tensor product ...
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53 views

Set notation: $U\otimes V\simeq V\otimes U$ with $V,U$ vector spaces

Let $U,V$ be finite-dimensional vector spaces over a common field and show that $U\otimes V\simeq V\otimes U$. Will someone explain what the notation $\simeq$ means in this case?
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26 views

$ch(M\otimes N)=chMchN$

Let L be a semi-simple Lie algebra and M be a finite-dimensional L-module. $M=\oplus _{\lambda \in H^*}M_{\lambda}$ be the weight space decomposition of M. Define $chM=\sum_{\lambda \in H^*} dimM_{\...
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1answer
16 views

How to expand $(\partial_\mu A^\mu)^2$

How would I expand the following: $$(\partial_\mu A^\mu)^2 \tag{1}$$ My understanding of it makes me think it would be as simple as: $$(\partial_\mu A^\mu)(\partial_\mu A^\mu)\tag{2}$$ but I ...
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1answer
92 views

Universal properties of tensor product of Lie algebra representations.

Let $\mathfrak{g}$ be a Lie algebra, and $V,W$ be two $\mathfrak{g}$ modules. Then one can define a $\mathfrak{g}$ module structure on $V\bigotimes W$ by: $x\cdot (v\otimes w)=(x\cdot v)\otimes w+v\...
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51 views

Describe prime ideals and Krull dimension of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$

I want to describe the prime ideals of $\overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}$, where $\overline{\mathbb{Q}}$ denotes the integral closure of $\mathbb{Q}$ in $\mathbb{C}$, ...
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A variation in the construction of the tensor product of modules

Let $A$ be a ring, $E$ a right $A$-module and $F$ a left $A$-module. Consider the free $\mathbf{Z}$-module $\mathbf{Z}^{(E\times F)}$ which comes with the injective canonical mapping $\phi:E\times F\...
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Why this flatness condition is needed? (Serge lang's Algebra)(p.618, chapter 16, proposition 3.7)

[My Question] Why "$F$ is flat is needed in this proof? [My attempt] Let $f : \mathcal{a} \otimes F \rightarrow \mathcal{a}F $ be defined by $f(\sum_i x_i \otimes b_i) = \sum_ix_i b_i$ . (By ...
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37 views

Does matrix multiplication work inside of tensors?

I have the matrix M $$ M=\begin{bmatrix} a&-b&0&0 \\ b&a&0&0 \\ 0&0&a&b \\ 0&0&-b&a \end{bmatrix} $$ which I would like to split into the tensor product ...
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40 views

Base change along a separable extension.

Let $L/K$ be a separable field extension of degree $n>1$. Is it true that $L\otimes_{K}L=L^{n}$ as $K$-algebras? My initial guess is yes, but I have no idea how to prove it.
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15 views

Is the canonical isomorphism $\mathbb R^m\otimes\mathbb R^n\to\mathbb R^{m\times n}$ isometric with respect to the Frobenius norm?

Let $p\in\mathbb N$ and $n_1,\ldots,n_p\in\mathbb N$. Assume the tensor product space $\bigotimes_{i=1}^p\mathbb R^{n_i}$ is equipped with the unique inner product $\langle\;\cdot\;,\;\cdot\;\rangle_{\...
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How does contraction apply to rank-1 tensors?

I'm just learning about tensors and the tensor product, and have been given examples for tensors such as $a \equiv x_1^2+x_2^2$ being one because it is the contraction of the tensorproduct of $\{x_i\}$...
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1answer
35 views

Monoidal adjunction whose right-adjoint functor has structure morphisms which are epimorphisms

Let $(\mathbf{C},\otimes,1)$ and $(\mathbf{D},*,e)$ be monoidal categories and let $L:\mathbf{C}\rightarrow \mathbf{D}$ and $R:\mathbf{D}\rightarrow \mathbf{C}$ be functors. Suppose that there exists ...
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46 views

Scalar restriction of bilinear maps

Let $R$ be a ring (commutative and with unity), $S\subset R$ be a subring. Consider three $R$-modules $M$, $N$ and $Z$. Let $\operatorname{Hom}_R(M\otimes_RN;Z)$ be the R-module of $R$-bilinear maps ...
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81 views

If $G$ is an Abelian Group of rank $r$ then $G\otimes_\mathbb{Z}\mathbb{Q}$ is isomorphic to $\mathbb{Q}^r$

So I'm trying to prove that if $G$ is an Abelian Group of rank $r$ (As $\mathbb{Z}$-module) then $G\otimes_\mathbb{Z}\mathbb{Q}$ is isomorphic to $\mathbb{Q}^r.$ Using the results I know about ...
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28 views

Showing well-definedness of addition and scalar multiplication for tensor product

Wikipedia defines the tensor product $V\otimes W$ of two vector spaces $V$, $W$ on a common field as partition $F(V\times W)~/ \sim$, not as a quotient space, so there is no a priori an obvious reason ...
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47 views

Non-zero element in exterior power

Let $R=\mathbb{Z}[\sqrt{5}]$ and consider $I=(2,\, 1+\sqrt{5})$ as an $R$-module. I'm struggling to prove that the element $2\wedge (1+\sqrt{5})$ of the exterior power $\Lambda^2(I)$ is non-zero. I ...
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1answer
26 views

Find orthonormal $\{b_i\}$ such that $\mathcal{A} = \sum_i \sum_j \lambda_i \mu_j b_i b_j^T$

Suppose $\{b_i\}_{i = 1}^d \subset \mathbb{R}^n$ is an orthonormal set of vectors, $d \leq n$, and assume that a matrix $\mathcal{A} \in \mathbb{R}^{n \times n}$ has the following form: $$\mathcal{A} ...
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1answer
10 views

Tensor contraction exercsie

I am working on a "tensor gymnastics" exercise, and have arrived at the following line to simplify: $\delta_{ik} y^{i} X_{ij}$ where $\delta_{ik}$ is the Kroenecker delta. Does this simplify to: $y^...
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11 views

How to make compact this Kronecker product sum

I would like to exploit some polynomial (matrix) property to make the following expression more readable and compact: $$ \sum_{k=0}^{K}h_{k}\left[\sum_{i,j=0}^{N} c_{ij} (A^{i} \otimes B^{j})\right]^...
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2answers
60 views

$\mathbb Z [x]$ module structure on $\mathbb Z$

For each integer $n \in \mathbb Z$, define the ring homomorphism $$φ_n :\mathbb Z [x]\to \mathbb Z, \ \ φ_n(f)=f(n).$$ This provides a $\mathbb Z[x]$-module structure on $\mathbb Z$ given by $$f ◦ a =...
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16 views

Wedge product of complexifications is complexification of wedge product

$V$ is a real finite dimensional vector space. Denote by $V_{\mathbb{C}}$ its complexification $V\otimes_{\mathbb{R}}\mathbb{C}$. I have already proved that $V_{\mathbb{C}}\otimes_{\mathbb{C}}V_{\...
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9 views

Complexification of real linear map and Hermitian metric

Consider $V$ a finite dimensional real vector space with Euclidean metric $\langle|\rangle$ $\langle|\rangle_{\mathbb{C}}$ the induced Hermitian metric on the complexification $V_{\mathbb{C}}=V\...
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46 views

Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? [duplicate]

Let $U$ and $V$ be modules over a commutative ring $K$. Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? I'm a differential geometer so I'm usually dealing with finite-...
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56 views

Irreducible decomposition of the representation $U\otimes\bar{U}$ of the unitary group

Short version Fix $d,t\geq 1$. Denote by $U(d)$ the set of unitary operators on $\mathbb{C}^d$. We can see $(\rho_t,\mathbb{C}^{\otimes 2t})$ is a representation of $U(d)$ where $\rho_t(U):=U\otimes\...
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16 views

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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20 views

Bivector to pseudovector mapping

I am studying antisymmetric tensors and currently reading a topic on pseudovectors. I understand that every bivector can be mapped to a corresponding pseudovector and vice versa but it is mentioned ...
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1answer
27 views

From vec-trick to matrix-trick for Kronecker products

for the vec-trick of the Kronecker product, we can write $$ \left(\mathbf{B}^{\top} \otimes \mathbf{A}\right) \operatorname{vec}(\mathbf{X})=\operatorname{vec}(\mathbf{A} \mathbf{X} \mathbf{B}). $$ ...
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1answer
24 views

Simplified tensor formula

A is a fourth-order tensor, B is a second-order tensor, and C is a second-order tensor $$\mathbf{A}:\mathbf{B}\cdot\mathbf{C}=[\quad ]:\mathbf{B}$$ $$\mathbf{B}\cdot\mathbf{C}=[\quad ]:\mathbf{B}$$ $$\...
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1answer
27 views

Dyadic Product Proof

How would I prove that $u \cdot (v \otimes w) = (u \cdot v) \otimes w$ where $u, v,$ and $w$ are first-order tensors ? The only concepts I've learned so far are properties of the dyadic product, ...
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1answer
53 views

$\operatorname{Hom}_{K}(K \otimes_k U, K \otimes_k V) \cong K \otimes_k \operatorname{Hom}_{k}(U, V) $

Let $K$ be a field extension of $k$, $U$ and $V$ finite dimensional (just in case) vector spaces over $k$. How do I construct an isomorphism like this? $\operatorname{Hom}_{K}(K \otimes_k U, K \...
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46 views

Tensor Product Properties

I am currently working my way through some notes and have got stuck proving a couple of tensor product properties. I have A,B,C,D as matrices and u,x,y as vectors, with a & b being constants. I ...
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1answer
102 views

kernel of Haagerup tensor product of maps

Haagerup tensor product $\otimes_{\rm h}$ is both injective and projective. Pisier, Gilles, Introduction to operator space theory, London Mathematical Society Lecture Note Series 294. Cambridge: ...
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1answer
50 views

When is $x \otimes y = y \otimes x$? [duplicate]

Let $X$ be a vector space and $x \otimes y \in X \otimes X$. Under which conditions is $x \otimes y = y \otimes x$? Does it nessecarily follow that $x = \lambda y$ for some $\lambda$ in the ...
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1answer
31 views

How does the “Alternating Operator” distribute in Tensors?

(I'm not sure that I even phrased the question correctly. I will explain more about this below.) Given a k-tensor $T$, we can define an alternating k-tensor $Alt(T)$ in the following way: where $\...
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How to simplify the curl of a second-order-tensor times a vector.>

I have the following equation: $$ \nabla \times (\sigma E) $$ where $\sigma$ is a second-order tensor and $E$ is a vector. I do have another equation I can substitute for $\nabla \times E$. How do ...
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1answer
26 views

Confusion regarding definition of tensor product of two vector spaces

Consider two vector spaces $E$ and $F$ over the same field $K$. Now to form the tensor product $E\otimes F$ we typically take a particular vector space $V_1$ and quotient it. The vector space $V_1$ is ...
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1answer
29 views

Can 2 contravariant vectors be multiplied by a tensor product?

How can you do the tensor product between the basis covectors $ \epsilon^i $ & $ \epsilon^j $ to result in $ \epsilon^i \otimes \epsilon^j $ when $ \epsilon^i $ & $ \epsilon^j $ have ...
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1answer
62 views

Cancelling the canonical module in tensor products

Let $R$ be Cohen-Macaulay local ring with the canonical module $\omega_R$ and let $M$ and $N$ be two finitely generated $R$-modules. Assume that $$ \omega_R \otimes_R M= \omega_R \otimes_R N $$ Can ...
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1answer
37 views

Infinite tensor product

Suppose that $X$ is an infinite set and $A$ is a unital $C^*$-algebra. The tensor product $\bigotimes_X A$ is defined to be the closed linear span of $\bigotimes_{x\in X }a_x$, where $a_x\in A$ for ...
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1answer
37 views

How to find matrix of antisymmetrization $\pi_A(g)$ where $g$ is the bilinear form $e^1\otimes e^1-e^1\otimes e^2+3e^2\otimes e^1+2e^2\otimes e^2$

Basis $M=\{(3,1),(2,1)\}$. I solved that the dual basis \begin{equation} M^*=\{e^1,e^2\}=\{(1,-2),(-1,3)\} \end{equation} Then I solved that matrix \begin{equation} g=\left(\begin{array}{cc}1&-...
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1answer
40 views

Geometric interpretation of total covariant derivative?

A connection $\nabla$ is said to be compatible with riemannian metric $g$ if $$\nabla_Z g(X,Y)=g(\nabla_Z X,Y) + g(X,\nabla_Z Y).$$ The total covariant derivative $(\nabla_Z g)(X,Y)$ can be ...
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12 views

How do you denote functions inherited by a product space?

Given a map $f:A\to B$, define the map $p_n(f):A^n\to B^n$ as $p_n(f)(a_1,...,a_i)=(f(a_1),...,f(a_i))$. Equivalently, you could say given $f$, $p_n(f)$ is such that $\mathrm{proj}_a\circ p_n(f)=f\...
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1answer
56 views

If $\mathbb Q \otimes_\mathbb Z \mathbb Q \cong \mathbb Q^\mathbb N$, why is $\mathbb Q \otimes_\mathbb Z \mathbb Q$ a $1$-dim $\mathbb Q$-v.s.

In Dummit & Foote, it is an exercise to show that $\mathbb Q \otimes_\mathbb Z \mathbb Q$ is a $1$-dimensional $\mathbb Q$-vector space. This is fairly easy: a $\mathbb Q$-basis for $\mathbb Q \...
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51 views

Simplest Examples for Tensors in Linear Algebra

In my attempt to gain some sort of understanding of tensor products (of vector spaces), and looking at the corresponding nLab entry (or Wikipedia's), I tried to consider a very basic example. ...
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1answer
34 views

Homogeneous Components of the Homogeneous Coordinate Ring of a Product of Projective Varieties

Suppose $X \subset \mathbb P^n$ and $Y \subset \mathbb P^m$ are projective varieties, and let $S(X)$ and $S(Y)$ be their homogeneous coordinate rings. Consider the projective variety $X \times Y$ in $\...

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