# 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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### 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 ...
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### Function for unique volume element

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 ...
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### Exterior powers and choice

Under the assumption that any vector space has a basis (so under the assumption of the axiom of choice), we can prove the following algebraic statements : 1) If $\varphi:V\to W$ is an injective ...
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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 an analogue of spin/oscillator representation for the general linear Lie algebra?

(Work over complex numbers) Let $V$ be an orthogonal space. Let $Pin(V)$ be the double cover of the orthogonal group $O(V)$. Then $Pin(V)$ has a basic spin representation which we can think of as the ...
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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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### Number of Symmetric matrices of fix rank over finite fields

This might be a question that shouldn't be asked here. But I need some help. I want to count the number of $n\times n$ symmetric matrices over the finite field $\mathbb{F}_q$ and rank $r$. I found the ...
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### Simultaneous “orthonormalization” in $\mathbb{C}^4$

Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix. So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
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### An upper bound for skew symmetric rank 2 matrices

Earlier, I had asked a similar question but that was not the correct problem where I got stuck. After a few quick answer, I realized that and I apologize for that. Let $B_m$ be the space of all skew-...
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### Conditions on $\beta$ under which the trace pairing restricted to $\mathfrak{so}(V,\beta)$ is positive (negative) definite

Let $V$ be a finite dimensional vector space over $\mathbb{R}$. Let $$\left\langle\:,\:\right\rangle:\mbox{End}(V)\otimes\mbox{End}(V)\rightarrow \mathbb{R}$$ denote the ...
If $F:V\to W$ is a smooth at $a\in V$ function between finite-dimensional vector spaces over $\mathbb{R}$, then we have $$F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder}, ... 1answer 93 views ### 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 ... 1answer 727 views ### 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 ... 1answer 577 views ### Tensor rank of anti-symmetric tensor Let V be a vector space of dimension n. Let us consider V^{\otimes n}=V\otimes V \ldots \otimes V. This vector space contains one dimentional vector space \wedge^n V. My question is does it ... 1answer 444 views ### 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 ... 1answer 111 views ### 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 ... 0answers 51 views ### 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)... 1answer 335 views ### Homogeneous polynomials and symmetric binary forms Let f\in k[x_0,...,x_n]_d be a degree d homogeneous polynomial in n+1 variables. Is there a way to associate to f a form g(y_1,...,y_m) which is symmetric in the sets of binary variables ... 2answers 411 views ### (Efficient) computation of symmetric powers of square matrices I'm looking for software that can compute symmetric powers of medium-size square (say rational, 100 by 100) matrices, and ideally can do so efficiently if the matrix is sparse enough. I haven't found ... 0answers 60 views ### Multilinear maps that preserve unitarity Let M_1, M_2, M_3 be spaces of square complex matrices, respectively acting on finite-dimensional Hilbert spaces V_1, V_2, and V_3 = V_1 \otimes V_2. Consider bilinear maps$$\phi: M_1 \times ...
Suppose I have a vector space with a tensor product structure $H = V^{\otimes m}$ and suppose I have some traceless Hermitian operator $A: V \otimes V \rightarrow V \otimes V$. Denote by $A_{i,j}$ ...