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Questions tagged [tensor-products]

Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their purely abstract incarnation (monoidal categories). To make the intent clear, this tag should only be accompanied by relevant other tags specifying the context.

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Help Computing Jacobians with Reshape/Tensors

I have the following variables in my problem statement, where I have the following matrices $V_{t}$ - [18540 x 3] 3D points $Tr$ - [62 x 12] 62 Transformation Matrices $W$ - [18540 x 62] ...
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1answer
36 views

Derivative w.r.t. x of Matrix Product $A(x)B(x)$

If $A(x)$ was a row vector and $B(x)$ was a column vector, below is true $$ \frac{d(A(x)B(x))}{dx}=\frac{dA(x)}{dx}B(x)+\frac{dB^t(x)}{dx}A(x)^t. $$ If you take derivative of $d_1\times d_2$ matrix ...
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1answer
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How to find the additive inverse and additive identity of an element of the tensor-product vector space?

The 8 axioms of holding for a vector space defined in Chapter 5 of the book Robertson's Basic Linear Algebra are easily checked for tensor product expect for existence of (unique?) additive inverse ...
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119 views

Torsion-free abelian groups, tensor product and $p$-adic integers

I'm studying torsion-free abelian groups and I know (see Fuchs, "Infinite Abelian Groups", vol. $2$, pp $154$) that, if $\mathbb{Z}_p$ is the set of $p$- adic integers and $\mathbb{Z}_{(p)}$ denotes ...
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Tensor product of function spaces: does $\widehat{\bigotimes}_{\pi;1\leq k\leq n}C(X_{k})^{\prime}\cong C(\prod_{k}X)^{\prime}$ hold?

Let $n\in\mathbb{N}$ with $n\geq 2$. Let $X_{k}$ be (non-empty) compact Hausdorff spaces for $1\leq k\leq n$. Let $X:=\prod_{k}X_{k}$ and let $\pi_{k}:X\longrightarrow$ be the projections. It is a ...
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A difficulty in understanding a part of the definition of tensor product.

when I searched about the definition of Tensor product of modules on Wikipedia I found the following sentence: "In mathematics, the tensor product of modules is a construction that allows arguments ...
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2answers
26 views

A difficulty in understanding the proof of distributivity of tensor products over direct sums for modules.

Here is the proof: But I do not understand the following: 1-why the function needed to be bilinear to use the universal property? 2- what is he doing starting from the paragraph that starts with ...
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1answer
51 views

A difficulty in understanding the universal property of modules.

The property is given below ( from Dummit & Foote) but I have a difficulty in understanding why it is universal property and what is its importance or when usually we use it?and why the function ...
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1answer
39 views

Weighted inner product with arbitrary matrix?

An inner product can be written in Hermitian form $$ \langle x,y \rangle = y^*Mx $$ that requires $M$ to be a Hermitian positive definite matrix. I have read that using Hermitian positive definite ...
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Basic resources for learning tensors

What are some "easy" resources /intro to tensors? (tensor products etc.). I am taking intro to QM right now and the professor is just telling us how to do basic tensor products on kets, operators ...
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CNOT quantum gate using tensor products

In textbook, it states that the CNOT gate with the X gate applied on second qubit is \begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&...
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1answer
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Why does $D\pi^i(p)(v)=v^i$?

There's an equality that I don't understand from Spivak' book "Calculus on Manifolds" (p. 89). We define $$ df(p)(v_p)=Df(p)(v) $$ Let us consider in particular the 1-forms $d\pi^i$. [...] ...
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31 views

Tensor product identities

I have to prove the following: $$S(a⊗b) = (Sa)⊗b$$ $$(a⊗b)S=a⊗(S^Tb)$$ $$(a⊗b)^T=b⊗a$$ $$tr (a⊗b) =a·b$$ $$(a⊗b) :S= (Sb)·a$$ I'm new with tensor calculus and my teacher ...
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1answer
22 views

Tensor contraction computation

I am currently trying to implement tensors as multidimensional arrays in C++, which is why i am asking this with regards to algorithmic computability. I would like to implement the contraction of two ...
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30 views

Infinite tensor product identity

Consider the Hilbert space $H = \mathbb{R}^2$ spanned by $\mathbf{e}_1$ and $\mathbf{e}_2$. A number of authors have considered the infinite tensor product $\mathcal{H} = \otimes_{i=1}^N H_i$, $N\to\...
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1answer
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What is the tensor product $\mathcal{O}_K \otimes_\mathbb{Q} \mathbb{R}$?

Let $m \in \mathbb{N}^*$. Denote the $m$-th cyclotomic polynomial by $\Phi(x)$ and a complex primitive $m$-th root of unity by $\omega$. Let $K = \mathbb{Q}[x]/\langle \Phi(x) \rangle$ (which is ...
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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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1answer
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M different particles moving in a graph of vertex order n

Let $G$ and $G'$ be a finite graph with vertex set of order $n$ and $n'$, respectively. For the Cartesian product of $G$ and $G'$, the resultant graph is an $\left(n \times n'\right)$ vertex graph. ...
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23 views

Definition of tensor product of two dual spaces of two given vector spaces

Let $X$ and $Y$ be two vector spaces over $\mathbb{C}$ and let $X^*$ and $Y^*$ be the algebraic duals of $X$ and $Y$, respectively. By definition, $X\otimes Y$ is the linear span of the set $\{eval_{(...
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Tensor and the $k$ times derivative of a smooth function $f$

I am currently taking an university analysis class. I learned yesterday in the class that (1) if $f: \mathbb{R^n} \rightarrow \mathbb{R}$ is smooth, then $D^kf: \mathbb{R^n} \rightarrow T^k(\mathbb{R^...
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1answer
60 views

Show that $M[x] \cong A[x] \otimes_{A} M.$

I'm trying to solve the problems in the book of Atiyah and MacDonald. I want to verify my solution to the problem 2.6. This is the exercise's statement: 2.6. For any $A$-module $M$, let $M[x]$ ...
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1answer
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A difficulty in understanding the indices in the matrix interpretation of the product of two representation.

I have a difficulty the indices in the two lines above the line starting with "Hence, $X$ transforms according to ..." , could anyone explain for me how the indices are located? Also I do not ...
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Comparing the representation $T \otimes T $ in terms of matrices and $T^2$

Interpret the representation $ T \otimes T$ in terms of matrices, and compare it with $T^2$. Could anyone give me a hint on how to solve this please ? EDIT: enter image description here enter ...
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1answer
47 views

Tensor Product of Hilbert Spaces: how to prove completeness

I'm studying tensor products of Hilbert spaces following the construction given in Folland's A Course in Abstract Harmonic Analysis. Let $\mathcal{H}_1$ and $\mathcal{H_2}$ two Hilbert spaces. ...
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1answer
16 views

deriving the Poisson equation from Navier stokes equation using tensor algebra.

I want to derive the poisson equation from the navier equations, but I'm not able to do this. I'm able to substitute the index form in place of the operators in the poisson equation and get back ...
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1answer
21 views

Prove commutation with tensor product

Say I have $\left|1,\psi\right>$ which is a vector in one representation of $\mathcal L$, where $\mathcal L$ is a Lie algebra and $\left|2,\phi\right>$ which is a vector in another ...
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65 views

Is there a natural way to view $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d$ as a subspace of $\bigwedge^k_{\mathbb{R}}\mathbb{C}^d$

Does there exists an $\mathbb R$-linear embedding $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d \to \bigwedge^k_{\mathbb{R}}\mathbb{C}^d$ which maps decomposable tensors to decomposable tensors? (The ...
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1answer
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Tensor Product of Quotient.

Let $K$ be a field and $L/K$ a field extension. Suppose $A$ is a $K$-algebra and $I$ is an ideal. I want to show that $$ (A/I\otimes_K L) \to (A\otimes_K L)/(I\otimes_K L)$$ So i define a map $$f: A\...
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1answer
40 views

How to construct a $k[G]$ isomorphism $W^G \otimes V \cong (W \otimes V_H)^G$.

Let $V,W$ be two $k$-vector spaces of finite dimension. Let $G$ be a group and $H \leq G$ a subgroup, with $[G:H]=n$. Let $\{x_1,\dots x_n\}$ be a right trasversal for $G$ on $H$ and assume that $V$ ...
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1answer
48 views

Isomorphism between tensor product of vector fields and their dual.

Consider two finite dimensional vector spaces $V_1,V_2$ and their duals denoted by $V_1^{*},V_{2}^{*}$. I am working on a problem that is asking me to prove a generalized version of the below, but I ...
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1answer
51 views

Two different definitions of tensor product space?

I'm currently taking a course on the mathematical foundations of QM and we're formalizing tensor products. The definition I'm used to which was taught in my differential geometry course seems to be ...
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Explicit form of elements of tensor products

Let $V$ and $W$ be two vector spaces over $\mathbb{F}$. We know that their tensor product $V \otimes W$ is also a vector space over $\mathbb{F}$. I am wondering how would look like the elements inside ...
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2answers
140 views

Covariant derivative of a symmetric tensor

Assume that a symmetric $(0,2)$ satisfies $$\nabla_iT_{jk}+\nabla_jT_{ik}+\nabla_kT_{ji}=0$$ where $T=T_i^i$ is constant and $\nabla_jT_{ik}\ne 0$. What are the values of the constants $a,b,c$ such ...
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what would be its solution in discrete as well as continous time

If I have an equation like ( assume everything as its suitable dimension and all are real matrices, $p,q$ are real constants) $(1)$ $x(k+1)= pf(x(k))+q(A\otimes B)x(k)$ $(2) \dot x(t)=pf(x(t))+q(A\...
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$R$ has an identity and $D$ is a projective unitary right $R$-module then sequence of corresponding tensor products is a short exact sequence

$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is a short exact sequence of left $R$-modules and $D$ a right $R$ module. $0 \rightarrow D \otimes_R A \xrightarrow{1_D \otimes\ f} D \...
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1answer
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How do I calculate an x with a circle in it and a variable raised to -T?

I am trying to design a decoupler to break a MIMO system down to a distributed SISO system for control. I am reading a paper on how to do that, but it is telling me to compute some strange stuff that ...
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1answer
35 views

$(A \otimes C) \oplus (B \otimes C) \cong (A \oplus B) \otimes C$

Let $A,B,C$ be R-modules. I know that _$\otimes C$ is an additive functor so $(A \otimes C) \oplus (B \otimes C) \cong (A \oplus B) \otimes C$ but what should the isomorphism send $(a \otimes c,b \...
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Higher moments of linear regression residuals?

Background In the following linear regression with i.i.d $\epsilon_i$ $(i = 1, \cdots, n)$ with mean 0 finite variance $\sigma^2$, \begin{align*} Y_i = X_i^\intercal\beta + \epsilon_i \end{align*} ...
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Dimension of relative tangent space

Let $k$ be an arbitrary field, $X$ be a $k$-Scheme locally of finite type, $x \in X$ a closed point and $\kappa(x)$ its residue field. Question: Is the dimension of of the tangent space $T_x X$ ...
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1answer
21 views

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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A special construction in $\mathbb{R}^n$

I have a question from my professor' notes. We defined $\Lambda^k(V)$ as the set of all $k$-Tensor' forms (multilinear transformations), $\omega$, which fulfuill $\omega(v_1,...,v_i,v_j,...,v_k)=-\...
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1answer
37 views

If $V$ is free, show that $f$ is surjective.

Let $R$ be a commutative ring with $1$. Let $V$ and $W$ be $R$-modules. a) Exhibit a canonical $R$-linear map $f: V^* \otimes V \to R $ b) If $V$ is free, show that $f$ is surjective. Now for the ...
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1answer
58 views

Coproduct of abelian categories

I know that there is a product in the category of small categories. I think this product is also the product in the category of pre-additive, or triangulated categories. There is also a coproduct of ...
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2answers
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Tensor square of a torsion-free module ($R$ domain)

Linked to my comment there. Let $M$ be a torsion-free module over a domain $R$. Is $M\otimes_R M$ torsion-free ? If not, a counterexample would be appreciated.
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Induced inner product on tensor powers.

Let $V$ be a real or complex inner product space with inner product $\left\langle \cdotp,\cdotp\right\rangle$. For $\otimes ^kV$ define $\left\langle \cdotp,\cdotp\right\rangle_k$ by $$\left\langle ...
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1answer
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Is the “tensoring map” from $\mathcal{H}_1 \times \mathcal{H}_2$ to $\mathcal{H}_1 \otimes \mathcal{H}_2$ a continuous map?

Let $\mathcal{H}_1$ and $\mathcal{H_2}$ two Hilbert spaces. Construct the tensor product $\mathcal{H}_1 \otimes \mathcal{H}_2$ as the set of all bounded antilinear operators from $\mathcal{H_2}$ to $\...
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Number field tensor $\mathbb{Q}$ isomorphisms. [closed]

Let $\mathbb{K}$ be a number field, i.e. a finite extension of $\mathbb{Q}$. Consider the tensor product $\mathbb{K} \otimes \mathbb{Q}$. I am not familiar with tensor products, so I have these ...
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Tensor Products for Bilinear Optimization?

I'm not very familiar with tensor products, but I recently read in a math textbook that " Tensor products are very important in algebra. They reduce the study of bilinear maps to the study of linear ...
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1answer
47 views

How can we define a covariant $k$-tensor this way if tensors are already defined by the tensor product?

From Lee's Intro to Smooth Manifolds: If we take $k$ covectors, $\varepsilon^{i_1}, \dots, \varepsilon^{i_k},$ then the tensor product is defined by $$\varepsilon^{i_1}\otimes \cdots \otimes \...
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1answer
51 views

Verapoulous Algebra $C(K) \mathbin{\hat\otimes} C(L)$ is a subalgebra of $C(K\times L)$?

Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. ...