# 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.

2,655 questions
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### 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: $T = e^1\oplus e^2 - e^2\oplus e^1 + 2e^2 \oplus e^2$ ...
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### 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?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 $$S\otimes_R \mathrm{Hom}_R(M,N) \simeq \mathrm{Hom}_S(S\otimes_R M, S\otimes_R N)$$ ...
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### 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 ...
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### 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 ...
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### 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 $(m',n)=(rm,rn')$ ...
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### 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 ...