# Questions tagged [multilinear-algebra]

For questions about the extension of linear algebra to multilinear transformations of vector spaces.

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### A perfect pairing with respect to the symmetric product

Let $V$ be a finite dimensional vector space endowed with a non-degenerated symmetric quadratic form $q$. Let $n\geq 1$ be a positive integer and $I_n$ the ideal of $\mathrm{Sym}^*V$ generated by ...
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### Coderivative of symmetric $2$-tensor

This is taken from a paper by M. Berger: Here $(X,g)$ is a Riemannian manifold and $h$ is a symmetric $2$-tensor on $X$. Could you help me to understand this definition, please? What does $\nabla^k$ ...
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### What is the generalization of Pascal's formula for multi indices?

I am taking a course in PDE and trying to get use to these notations. If I take 3 vectors with nonnegative integers components (n components) that are denoted by $\alpha$,$\omega$,$\gamma$. Is it ...
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### Is the alternatization of a m-form on a m-dimensional vector space non-zero if there exists a basis on which the m-form is non-zero?

Let $V$ be a $m$-dimensional real vector space, and let $e_1,\ldots, e_m$ be a basis such that the $m$-form $f$ has the property $$f(e_1,\ldots,e_m)\neq0$$Is $\operatorname{Alt}(f)\neq0$? Or does ...
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### Objects that generalizes Universal enveloping/Clifford/Weil algebra and correspondence to algebraic geometry.

There are evident similarities between Clifford, Weil and Universal Enveloping algebras. Each may be defined directly as a quotient of the tensor algebra $T(V)$ divided by an ideal generated by some ...
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### Defining an $A$-algebra structure using a basis and “constants of structure”

Let $A$ be a commutative ring and $E$ an $A$-module with a basis $(e_i)_{i\in I}$. Let $(\alpha_{ijk})_{(i,j,k)\in I\times I\times I}$ be a family of elements of $A$ such that, for $i,j\in I$, the ...
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### Are $(1, 0)$ tensors always vectors? (resolved)

An $(r, s)$ tensor $T$ is defined to be an element of the tensor product of a vector space and its dual: $$T \in T^r_sV := V^{\otimes r}\otimes V^{* \otimes s}.$$ However, when $V$ is finite ...
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### Single-factor CP decomposition optimization equivalency with block coordinate descent using the tensor power method

I am trying to understand the objective functions for CP decompositions using the tensor power method as discussed in this paper. In this approach, a series of rank-1 approximations are made using the ...
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### Clifford map is injective for $B=0$.

Here's my definition of a Clifford algebra: Definition: Let $B(\cdot,\cdot)$ be a symmetric bilinear form on a vector space $V$ over $\mathbb{K}$ and $Q$ its associated quadratic form. The Clifford ...
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### Finding the spectra of difficult parametric matrices

Preparing for entrance exams, I am in need of finding the spectra of the following matrices the most effectively. Anyone up for helping me find the best ways? Matrix 1 | Matrix 2 | Matrix 3 The ...
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### extrior product of a square matrix

I want to understand how one can find the exterior product of square matrices. I searched about it and I find a lot of complicated formulas and without any examples. On the other hand, I found this ...
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### Correct definition of scalar triple product

Consider orieneted euclidean space $\mathbb{R}^3$. A vector $v\in\mathbb{R}^3$ determines a $1$-form $\omega_v^1$ where $\omega_v^1(w)=\langle v,w\rangle$, where the brackets denotes the equipped ...
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### The optimal solution of tensor Tucker decomposition problem

I have a question about tensor Tucker deocomposition. Recall that in Higher-Order Singular Value Decomposition (HOSVD), each factor matrices $U_i$ is obtained by taking the top $R_i$ left singular ...
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### Generalizing the geometric interpretation of dot product to simple $k$-vectors

Background: For $u, v \in \mathbb R^n$, the dot product $u \cdot v$ can be interpreted geometrically as follows: Its magnitude is the product of the lengths of $u$ and $\operatorname{proj}_{u} v$. ...
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### Specific example of the universal property of $V \otimes W$

I'm trying to familiarize myself with the definition of tensors, so I was wondering if I understood the definition in terms of the universal property. Consider a bilinear $B : V \times W \rightarrow U$...
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