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 ...

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22 views

Border rank of tensors

Can anyone help me find the rank and border rank of the following tensor: \begin{align} T=a_{11}\otimes b_{11}\otimes c_{11}+a_{12}\otimes b_{21}\otimes c_{11}+a_{11}\otimes b_{12}\otimes c_{12}\\ ...
2
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1answer
33 views

Littlewood Richardson rules for the orthogonal group SO(d)

I have a question related to tensor products of Young diagrams. More precisely, I know the Littlewood Richardson rules for the general linear group GL(d) and would like to know the restriction of ...
1
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1answer
54 views

Embedding tensor product of integral domains

Let $C$ be a subring of integral domains $A,B$ and let $C',A',B'$ denote their field of fractions respectively. Can we always embed $A\otimes_CB$ in $A'\otimes_{C'}B'$ by $a\otimes b\mapsto ...
2
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1answer
31 views

Tensors in matrix multiplication algorithms

Fast matrix multiplication algorithms, be it the Winograd and Coppersmith algorithm or any further improvement of it, extensively use tensors. In fact, the entire construction is based on tensor ...
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39 views

Decomposition of an unitary operator by simple operators

For quantum computation, it's well known that any unitary operator can be approximated with an arbitrary accuracy by simple operators, for example to approximate an unitary operator on n qubits by no ...
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2answers
65 views

Find trace of linear operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$ by permutation of the basis vectors. Suppose we know its eigenvalues ( some roots of unity ): ...
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1answer
31 views

What do tensors of second order map to?

On page 15 of James G. Simmonds book "A brief on Tensor Analysis" (chapter 1 of the first published edition), a second order tensor is described as an operator that sends vectors into vectors. On ...
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14 views

Permutation unitary in a tensor product

Given a matrix of the form $$ A = B_{1} \otimes B_{2} \otimes B_{3} \otimes... \otimes B_{n} $$ how can I find a matrix that gives me a permutation of , say, two of the elements: $$ A = B_{2} ...
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1answer
50 views

Find eigenvalues of operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$. Suppose we know its eigenvalues - $\lambda_1, \lambda_2, \ldots, \lambda_n.$ Now consider the ...
2
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2answers
42 views

Mathematical formalism for the “dot product” of three vectors

I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation: (x,y) = sum(x.*y). I'm interested in ...
4
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1answer
58 views

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$.

Show that $1\otimes (1,1,\ldots)\neq 0$ in $\mathbb{Q} \otimes_{\mathbb{Z}} \prod_{n=2}^{\infty} (\mathbb{Z}/n \mathbb{Z})$. Here's what I tried: If $1\otimes (1,1,\ldots)= 0$, then $1\otimes ...
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41 views

Converting tensor product from one coordinate to another

This is a long multi-steps question and I'm stuck at the last leg. I believe my question to be trivial but after 3 hrs of staring and trying all sort of methods (ridiculous ones even) I'm not getting ...
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0answers
35 views

Making sense of multiplication in tensor product

Let $\mathbb{Z} / p$ and $\mathbb{Z} / q$ be $\mathbb{Z} /pq$ modules where $p$ and $q$ are distinct primes. Then $\mathbb{Z}/p \times \mathbb{Z}/q$ is a $\mathbb{Z} / pq$ module Now let $$ i : ...
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1answer
35 views

Derivation of tensor components transformation in tangent space

Might anyone offer a derrivation? My attempt bellow ($ x_{i'}$ is counting through the transformed coordinates) $\displaystyle\frac{\partial }{\partial x_{i'}}= \displaystyle\frac{\partial ...
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2answers
52 views

If $f\otimes_\mathbb{Z}\mathbb{Z}/(p)\colon M\otimes_{\mathbb{Z}}\mathbb{Z}/(p)\to N\otimes_\mathbb{Z} \mathbb{Z}/(p)$ is onto for all $p$, $f$ onto?

This lemma is used in a theorem I'm reading, with no proof. Suppose $f\colon M\to N$ is a morphism of free, finitely generated $\mathbb{Z}$-modules. Then if $f\otimes_\mathbb{Z}\mathbb{Z}/(p)$ is ...
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0answers
11 views

Is basis for a Matrix field unique?

In general, when expressing a Matrix field, Q, in terms of it's local and dual basis (the tensor product between a a row vector and column vector), is such basis unique or are there infinitely many ...
3
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2answers
78 views

Paul Garret's Proof of the Cayley-Hamilton Theorem

I am trying to understand the proof of Cayley-Hamilton Theorem given in Paul Garrett's notes. We have a finite dimensional vector space $V$ over a field $k$ and are given a linear operator $T\in ...
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1answer
44 views

Representations of cartesian product $G$

We know for both representations of a locally compact group $G$, their tensor product is a representation of $G \times G$ (cartesian product of $G$ with self). Is each representation of it of this ...
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2answers
37 views

Notation Clarification: $M\odot N$ for von Neumann algebras $M$ and $N$

Given a Banach space $E$, $y\in E$, $\phi\in E^{*}$, I am led to the understanding that $y\odot \phi$ denotes the operator in $B(E)$ defined by $$x\mapsto \phi(x)y,\text{ for all } x\in E$$ Now I ...
3
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1answer
22 views

Symmetric tensor powers as tensors over symmetric group algebra

Let $V$ be a $k$-vector space and $V^{\otimes n}$ the $n$-fold tensor power of $V$ and let $\mathbb{S}_n$ be the symmetric group of an n-element set, with its signum representation denoted by ...
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1answer
28 views

To Show that $R/I\otimes_R R/J\cong R/(I+J)$

Let $R$ be a commutative ring with identity and $I$ and $J$ be ideals in $R$. Show that $R/I\otimes_R R/J\cong R/(I+J)$ as $R$-modules. This is what I tried. Define a map $f:R/I\times R/J\to ...
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0answers
15 views

Product of Symmetric and Antisymmetric Tensors

I would like to know if the following product is antisymmetric in 4 dimensions. All tensor elements are real valued. $${T_\mu}^{\nu}$$ T is a 4 dimensional symmetric tensor with mixed indices. ...
3
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1answer
48 views

Understanding the algebra of polynomials on a linear space

My advisor and I are working through a paper on partition functions, and we got to the following passage: Fix $n \in \mathbb N$ and let $W := ((\mathbb R^n)^{\otimes 3})^{C_3}$, where the $C_3$ ...
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31 views

Does $M \otimes_R N = 0$ for a non-unital ring $R$ if there are ideals $I,J \lhd R$ such that $MI+JN = 0$ and $I+J = R$?

Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$. This is very easy to prove if the ring is unital as you may write ...
2
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1answer
62 views

$d \times d \times d$ tensor rank vs $d \times d$ tensor rank

I am trying to understand rank of a $d \times d \times d$ tensor. The way that I understand the $d \times d$ case is that a rank $r$, $d \times d$ tensor is a tensor that can be written as the sum of ...
4
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2answers
174 views

Subspaces of a tensor product of vector spaces

What are the subspaces of tensor product of two vector spaces? Is every subspace of tensor product $V\otimes W$ of the form $V_1\otimes W_1$, where $V_1, W_1$ are subspaces of $V$ and $W$?
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44 views

Example of two modules M, N where the set of the $m\otimes n$ is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong ...
2
votes
1answer
48 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
2
votes
1answer
47 views

Why is this a bounded operator?

Let $\mathcal{H}$ be the Hilbert space $l^2(\mathbb{N})\otimes l^2(\mathbb{Z})$. I want to prove that the operator $T$ defined by $$T:=\sum_{k=1}^{\infty}{\sqrt{1-q^{2k}}e_{k-1,k}\otimes 1}$$ is a ...
2
votes
0answers
14 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
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25 views

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
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0answers
14 views

The tensor product of $ {L^{1}}(G) $ and a Banach space

Let $ G $ be a locally compact group and $ A $ a Banach space. It is known that the tensor product $ {L^{1}}(G) \otimes A $ is isometrically isomorphic to $ {L^{1}}(G,A) $. I need a proof of it.
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votes
2answers
101 views

Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?

I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in ...
2
votes
0answers
31 views

tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
1
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1answer
41 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
1
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1answer
29 views

Copower functor

Computing copowers and "tensoring with sets" often means the same thing. If a locally small category $\mathcal{C}$ has coproducts and if $S$ is a set then for any object $C\in\mathcal{C}$ the copower ...
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1answer
46 views

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
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1answer
41 views

What is $\mathrm{dim}(\mathrm{Sym}(\mathrm{Herm}(H)^{\otimes N})$?

The totally symmetric subspace of $(H^k)^{\otimes N}$, with $H^K$ a $k$-dimensional Hilbert space, has dimension $\binom{N+k-1}{k-1}$. But I now want to know the dimension of the totally symmetric ...
3
votes
1answer
44 views

Tensor product isomorphic to a free module

Is it true that if $R$ is a domain with quotient field $K$ and $M$ is a finitely generated torsion-free $R$-module then $M\otimes_R K$ is isomorphic with $K^n$ for some $n$? I know that the first is ...
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1answer
35 views

A question about 1.0.3 in Grothendieck's EGA

In (1.0.3), Grothendieck states that, given non-commutative rings $A$ and $B$, a homomorphism $\varphi : A \to B$, and a left ideal $\mathfrak{J}$ of $A$, the left ideal $B\mathfrak{J}$ of $B$ ...
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0answers
13 views

Solving a quadratic vector/tensor equation arising from connected Markov chains

I have a discrete-time finite-state aperiodic irreducible Markov chain, which is composed of $m$ identical component sub-chains. With probability $1-\mu$, in each time step each of these chains ...
3
votes
1answer
46 views

Basic Confusion About Tensor Products

Let $A$ and $B$ be subspaces of vector spaces $V$ and $W$ respectively. Given $a\in A$ and $b\in B$, there are two possible interpretations of $a\otimes b$: we can think of it as a member of ...
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0answers
11 views

Doubt regarding the definition of tensor product

I refer to the following image from "Atiyah-Macdonald", which describes the tensor product of modules. I don't understand how D forms a module. How exactly should I had two elements in D?
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1answer
29 views

Some calculations with skew forms and wedge product

I have some problems with the language of multilinear forms. I have to prove that if $dim(V)\le 3$, then every $\omega\in\Lambda^q(V^\ast)$ is such that $\omega\wedge\omega=0$. I consider the case ...
2
votes
0answers
38 views

Basis of a vector space using basis of tensor product and universal property

I already proved that if $\{u_i\}$ is a basis for $U$, $\{v_i\}$ basis for $V$, then $\{u_i \otimes v_j\}$ is a basis for $U \otimes V$. Now, I am trying to prove that if $g:U \times V \to W$ is ...
4
votes
1answer
27 views

Example tensor representation

Studying from Roman's Advanced Linear Algebra I got stuck with the following exercie, I want to find two vector spaces $U$ and $V$ and a nonzero vector $x\in U \otimes V$ that has at least two ...
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1answer
44 views

Example of a bilinear map whose image is not a subspace

I am looking for an example of a bilinear map $\tau:V \times V \to W$ whose image $im(\tau)=(\tau(u,v):u,v \in V)$ is not a subspace of $W$. I considered the tensor map $\tau:U \times V \to U \otimes ...
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1answer
45 views

Isomorphism between dual space and bilinear forms

Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ ...
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votes
1answer
30 views

How to compute tensor product between two functions

Let $f$ and $g$ are two functions defined on $[0,1]$ and taking values in a Hilbert space $H$. Then how to define and compute the tensor product between $f$ and $g$, namely $f\otimes g$? I know how ...
3
votes
1answer
59 views

Star operator in the simplest form

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual ...