# Questions tagged [multilinear-algebra]

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

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### The upper bounds on rank $2$ real matrices

Let $A_{n}(F)$ be the collection of all skew-symmetric matrices over the field $F$ ($\operatorname{char} F \neq 2$). Let M be a subspace of $A_{n}(F)$ such that all non zero elements have rank ...
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### Geometric characterisation of polynomials between normed spaces

Let $(X, \| \cdot \|_X)$ $(Y, \| \cdot \|_Y)$ be normed space. A function $f \colon X \to Y$ shall be called an $n$-th degree (single variable) polynomial ($n \in \mathbb{N}\cup \{ 0\})$ if there ...
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### Solving nonlinear differential multi-variable equation with block-matrices

Here is the problem: Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...
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### Map between irreducible representations in basis given by Young tableaux

Let $V$ be a $n$-dimensional complex vector space. Assume we have a $\mathbb C$-linear map $\varphi:\Gamma^{(a_1,\dots,a_n)}V\rightarrow \Gamma^{(b_1,\dots,b_n)}V$ between two irreducible ...
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### Define an inner product between p-blades so that 0= completely orthogonal and 1=completely overlapping for their subspaces

$\DeclareMathOperator\span{span}$Denote two $p$-blades $\nu=v_1\wedge \dots \wedge v_p$ and $\omega=w_1\wedge \dots \wedge w_p$ $\in \bigwedge^p X$, where $X$ is an inner product space. How to define ...
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### What is this SVD (called) with a singular value vector and U and V are tensors?

I am looking for information on a specific type of tensor/matrix decomposition which is quite similar to the SVD for matrices but does not look like the HOSVD since the core tensor is only a vector. ...
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Let $A,B$ be two symmetric tensors of same rank $m$; and orders $k$ and $\ell$, respectively. In particular, assume that $A,B$ admits the following structure: There exists $v_1,\dots,v_m\in\mathbb{R}^... 0answers 66 views ### Is there a way of separating symmetric multilinear forms? I have a very simple question which seemingly falls into the category of multilinear algebra. I need this concept in one of my research papers, but unfortunately did not learn it before. Let me state ... 1answer 252 views ### Sketching Frobenius norm of a tensor with a rank-1 random tensor Let$A\in\mathbb{R}^{n^k}$be a$k$-dimensional tensor with$n$elements along each dimension. Moreover suppose$u_1,u_2,\dots,u_k\sim\text{Unif}(\pm1)^n$are$n$dimensional vectors with each of ... 1answer 514 views ### Exterior derivative independence from coordinate systems In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ... 0answers 624 views ### Normal multivariate orthant probabilities (Previously I posted a similar question on math.SE, hoping that this question would have an easy answer. As the question appears hard, I am hoping I can perhaps get more feedback here.) Let$\mathbf{...
This is an issue that I'm am trying to solve for a fine-tuning measure in particle physics, but it is purely mathematical. Consider three vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$. I would like a ...
### Finite dimensional commutative algebras containing infinitely many nilpotents whose $d$-way products are nonzero
I'm interested in the following strange question: for some $d > 1$, what is the minimum dimension of a commutative $\mathbb{C}$-algebra containing infinitely many elements that square to zero, but ...