# Questions tagged [multilinear-algebra]

Tensors, multivectors, wedge products, multilinear maps, exterior (Grassmann) algebras.

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### Computational complexity of higher order orthogonal iteration for Tucker decompositions

I am currently doing background reading for my Masters Thesis. I am working with tensor decompositions, where by tensor I simply mean a multi-dimensional array. The aim of my masters project is to ...
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### Efficient way to express a symmetric tensor in terms of rank one elements

Let $$P(\mathbf{x})=P(x_1, \ldots, x_n)=x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}$$ be a homogenous polynomial of degree $k=k_1+\cdots+k_n$. It follows from a standard polarization identity (see for ...
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### A symmetric bilinear form and a Plücker identity

It turns out that a special case of something I'm working on gives, as a corollary, a rather 19th-century-looking elementary statement about the rank of a certain symmetric matrix. I thought I would ...
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### Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
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### Is there a generalization of eigenvalues and eigenvectors to tensors?

Two perhaps ill-posed or just silly questions: Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...
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### Looking for a tractable algorithm or formula for the determinant of a tensor

It is possible to define the determinant of a tensor. We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map. If $T:\{1,....,n\}^m\to \mathbb C$ ...
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### Sum of certain decomposable elements

Let $V$ be be a vector space of dimension $m$ over any field and $\ell\leq m$ be a positive integer. Let $\omega_1,\ldots,\omega_r \in\bigwedge^\ell V$ are linearly independent, completely ...
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### Tracing the word “form”

Today the word form can refer to (at least) three different kinds of mathematical object: A homogeneous polynomial. This was apparently started by Gauss (1801), renaming what others had called ...
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### Waring rank vs tensor rank of symmetric tensors?

Suppose we work in an algebraically closed field. Then, do the Waring rank (symmetric tensor rank) and tensor rank of a symmetric tensor coincide in general? Recall that tensor rank is rank with ...
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### Non alternative $k$-linear maps vanishing on $\sum x_i=0$

Assume that $V$ is a finite dimensional real vector space of dimension $n$. Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple ...
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### Maximum number of matrices satisfying given rank conditions

Assume that we have $2k$ matrices $S_1,\ldots,S_k$ and $\Phi_1,\ldots,\Phi_k$ over some finite field $F$ such that (i) $S_i\in F^{l/2\times l}$ and $\dim S_i=l/2$ for any $i\in\{1,\ldots,k\}$; (ii)...
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### Subgroups of the tensor product $A\otimes A$
I have this problem about subgroups of the tensor product of an abelian group $A$ with itself which arises from a complete different setting. I fell into this question studying quandles and quandle ...