# Questions tagged [multilinear-algebra]

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

128
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### Orbit closures of real symmetric bilinear forms

Let $\alpha$ and $\beta$ be two real symmetric bilinear forms in $\operatorname{sym}(\mathbb{R}^{n})$, with signatures $(p_{\alpha},n_{\alpha},z_{\alpha})$ and $(p_{\beta},n_{\beta},z_{\beta})$.
...

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172 views

### Why some operations on tensors don't give a tensor? [closed]

I asked the following question on math.stackexchange but no one seemed to have an authorative answer so I'm posting here hoping that experts will see it.
The gradient is a tensor $\nabla f:\mathbf{V} \...

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55 views

### Is there an analog of polarization for skew-symmetric forms?

This question might be too lightweight here but on math.SE it did not receive any feedback since May 2, so...
Polarization works both ways. Not only can you represent any homogeneous polynomial $f$ of ...

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109 views

### Subgroup of $PGL(n(n-1)/2, \mathbb K)$ preserving the grassmannian $Gr(2, n)$

How can we determine the subgroup of $PGL(\wedge^2 \bar{\mathbb Q}^n)$ which preserves the grassmamnnian $Gr(2, n)$ embedded as a projective variety in $\wedge^2(\bar{\mathbb Q}^n)$ via the Plucker ...

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435 views

### What is the role of topology on infinite dimensional exterior algebras?

Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows.
Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...

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**1**answer

50 views

### Analytical decomposed form of a specific traceless symmetric tensor

Assume an m-way tensor $\mathcal{Z}$.
$\mathcal{Z}_{p_1 p_2 ... p_m} = 0$ if any different indices match
and $\mathcal{Z}_{p_1 p_2 ... p_m} = 1$ otherwise.
It is a symmetric tensor. Now if it is 2-...

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45 views

### Characterization of all-orthogonal tensors

In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...

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79 views

### A real system of bilinear equations with $2n$ unknown and equations

I have the following system of $2n$ bilinear equations, for a square invertible matrix $A \in \mathbb{R}_{n \times n}$, and $2n$ unknowns organized in vectors $x,y \in \mathbb{R}^n$:
$$
diag(y) A x = ...

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127 views

### action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$.
Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via
$$\pi(e_i \wedge e_j) ...

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227 views

### Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?

It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...

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80 views

### Finding a specific solution to $X^T\Sigma X = D$

I'm looking to solve for a specific $X$ in the following equation:
$$X^T\Sigma X = D,$$
where $\Sigma \succ 0$, $D$ is a diagonal matrix with strictly positive entries, and all matrices are square. It ...

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97 views

### Multilinear Morse functions on the n-torus

Consider the $n$-dimensional Torus $T^n = \prod_{i=1}^n S^1$ as a subset of $\mathbb R^{2n} = \prod_{i=1}^n \mathbb R^2$ in the standard way.
Is it true that a generic $n$-multilinear functional on $\...

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**1**answer

371 views

### Homomorphism induced by the second exterior power of a linear map

Consider the map from $M(n, \mathbb Z) \rightarrow M(\binom{n}{2}, \mathbb Z)$ taking a matrix A to its second compound, i.e, $\bigwedge^2 A$.
Restricting this map to the invertible matrices we get a ...

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36 views

### 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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212 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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**1**answer

45 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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**1**answer

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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118 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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49 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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38 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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55 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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204 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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291 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 ...

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319 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{...

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54 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 ...

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300 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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280 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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49 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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2k 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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249 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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55 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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211 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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185 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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796 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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**1**answer

123 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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998 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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**2**answers

456 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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**1**answer

94 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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64 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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302 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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144 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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74 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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**1**answer

191 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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86 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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**1**answer

151 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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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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233 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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**1**answer

94 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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**1**answer

739 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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**1**answer

501 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 ...