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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1answer
194 views

Irreducible representations of $\mathrm{SL}_n(K)$, $K$ finite

Let $\mathrm{SL}_n/K$ ($K$ finite) be given with its natural action on an $n$-dimensional vector space $V/K$. Consider the action of $\mathrm{SL}_n$ on the $m$-fold tensor product $V\otimes \dotsc \...
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1answer
37 views

Expectation value of multilinear forms over independent Gaussian vectors

Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable: \begin{align*} X=A(g_1,\ldots,g_n,g_1,\...
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5answers
2k views

Generalization of the polarisation formula for symmetric bilinear forms to symmetric multilinear forms

Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vector space and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$. The Polarisation Formula ...
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1answer
2k views

Why does this matrix have zero determinant?

This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
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0answers
107 views

Transformations of the cubic forms [closed]

Is there a way to understand whether there exist linear transformation that brings one cubic form of n variables to another form? In particular one of the examples I am interested in are two cubic ...
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0answers
42 views

Rank-1 decomposability of symmetric tensors

My question is about rank-1 decomposability of symmetric tensors over the reals. Let $v_1,\dots,v_n\in\mathbb{R}^d$ be vectors. Construct the object: $$ V=\sum_{j=1}^n \underbrace{v_j\otimes v_j\...
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1answer
172 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 ...
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0answers
34 views

On symmetric tensors with same rank, different orders

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}^...
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49 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 ...
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3answers
2k views

A nice necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure

Let $$ A = \begin{pmatrix} \sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\ -a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\ \vdots & \vdots & \ddots & \...
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1answer
196 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 ...
2
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0answers
173 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{...
7
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1answer
277 views

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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0answers
53 views

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 ...
7
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1answer
256 views

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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1answer
199 views

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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1answer
1k views

Why is the standard definition of a $(p, q)$-tensor so bizarre?

At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows. Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
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1answer
1k views

Determinant of exterior power

Suppose $A$ is a $n$ times $n$ matrix. What is the determinant of the $i$-th exterior power of $A$, in terms of determinant of $A$?
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42 views

Integrating over some domain of the Stiefel manifold to analyze its support

Define the square $d$ dimensional Stiefel manifold as $$V_{d} = \{ R \in \mathbb{R}^{d \times d} : R ^\top R = I_d \} .$$ How does one integrate on this manifold over a domain defined as $\{ R \in V_{...
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1answer
238 views

Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This is a cross-post. Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...
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0answers
50 views

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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1answer
173 views

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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1answer
374 views

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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1answer
116 views

Non-negativity condition for special quartic

I know that a necessary and sufficient condition for the positivity of a quartic polynomial of many variables is in general difficult. I have a somewhat special case, maybe here more can be said. Let $...
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1answer
1k views

Exact sequences of bundles on Grassmannians

We're looking for a large set of exact sequences of vector bundles on Grassmannians. Here's the set up: $V$ and $Q$ are complex vector spaces of dimensions $d$ and $r$ respectively $(d\geq r)$, and ...
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2answers
452 views

A kind of “Curvature tensor” for higher dimensional tensors

I begin my question with a multilinear question then I will consider two local smooth analogies: Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...
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2answers
646 views

Clifford algebras as deformations of exterior algebras

$\def\Cl{\mathcal C\ell} \def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$ I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question. A well ...
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0answers
63 views

Points on Sphere whose image, under symmetric positive definite matrix, is contained in cube

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a symmetric, positive definite matrix and let $\mu_r$ denote surface measure on the sphere in $\mathbb{R}^n$ with radius $r$. Let $$ R = \{x \in \mathbb{R}^...
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1answer
411 views

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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1answer
91 views

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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1answer
240 views

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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0answers
121 views

Number of Symmetric matrices

Let $S_m(q)$ denote the space of all $m\times m$ symmetric matrices over the finite field $\mathbb{F}_q$ of size $q$. What is the number of matrices $A=(a_{ij})\in S_m(q)$ of rank at most $3$ and $a_{...
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0answers
70 views

Maximum Number of Skew-Symmetric matrices

I want to count the maximum number of rank 2 matrices in a space of certain dimension but I am stuck at some point. Any help/ suggestions are appreciated. Here is the question. Let $\mathbb{M}_m$ be ...
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1answer
161 views

Is the Waring rank homogeneous polynomials sub-multiplicative?

For a homogeneous degree $d$ polynomial $P$, the symmetric or Waring rank $W(P)$ is the minimum $r$ such that $P = \sum_{j=1}^r l_j^d$, where $l_j$s are linear forms. Now, is the Waring rank sub-...
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0answers
85 views

Number of vectors such that the projection is decomposable

Let $V$ be a vector space of dimension $n\geq 6$ over the finite field $\mathbb{F}_q$. Let $\omega\in\bigwedge^{n-3}V$ be a nonzero element. Define the annihilator subspace of $\omega$ by $V_\omega=\{...
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3answers
1k views

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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1answer
133 views

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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1answer
160 views

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 \begin{equation} \left\langle\:,\:\right\rangle:\mbox{End}(V)\otimes\mbox{End}(V)\rightarrow \mathbb{R}\end{equation} denote the ...
4
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0answers
216 views

Formal multidimensional Taylor series expansion over commutative rings

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}, ...
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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 ...
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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 ...
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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 ...
5
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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 ...
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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 ...
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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)...
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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 $...
5
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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 ...
3
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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 ...
2
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0answers
95 views

Spectrum of a special sum of matrices

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 $A_{i,j}$ ...