# Questions tagged [tensor-decomposition]

For questions related to tensor decomposition. A tensor decomposition is any scheme for expressing a tensor as a sequence of elementary operations acting on other, often simpler tensors.

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### Higher order Tensor computations

I have a $D$-way tensor H of dimension $I \times I \times \dots \times I$ ($D$ times), that represent the coefficients of a polynomial. For better understanding, I provided an image of 3-way tensor ...
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### How to calculate $\text{End}(V^{\otimes n})$

Let $\mathfrak g$ be a complex semisimple Lie algebra, and $V$ the fundamental $\mathfrak g$-module. Then we can decompose $V^{\otimes n}$ into the direct sum of irreducibles. For example, in the case ...
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### fundamental questions regarding approximations of tensors

My situation: I'm an absolute beginner regarding tensors and Ive currently working on a manuscript about a format which approximate / represent tensors of possibly high order. But before I can start ...
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### Decomposing the all $1$s matrix in tensor products

Consider the following matrices $$I = \begin{pmatrix}1& 0\\ 0 & 1\end{pmatrix},\quad X = \begin{pmatrix}0& 1\\ 1 & 0\end{pmatrix}$$ Define the indexed quantity $P^n_k(I, X)$ to be ...
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### Are all $v \in \Lambda ^k V$ decomposable if $k > \frac{1}{2} \dim V$?

Update: It seems I proved only the $(\dim S)$-decomposibility of the sum of two decomposables, since $s_1 + s_2$ is not decomposable (which I'm not quite sure of). Original post: Let $V$ be any $n$-...
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### Decomposable tensors

A tensor $t \in V \otimes W$ is called decomposable if $t = v \otimes w$ for some $v \in V$ and $w \in W$. The set of decomposable tensors is the image of the map f : \mathbb{R}^2 \...
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### Relation between symmetric outer product decomposition and symmetric multilinear decomposition

Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is $$\mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k},$$ ...
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### How to build 2nd rank tensors from set of vectors?

Suppose we have 3 vectors: $A_i$, $B_i$, $C_i$, where $i=1,2,3$. How to build all possible dependent 2nd rank tensors based on these vectors? I believe, that at first we need to obtain all possible ...
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Let $\mathcal{B(H)}$ be the space of all bounded linear operators on the Hilbert space $\mathcal{H}$. Let $g \rightarrow \mathcal{U}_g$, where $\mathcal{U}_g (\cdot):= U(g) (\cdot) U(g)^\dagger$, be ...