Questions tagged [multilinear-algebra]

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

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4
votes
1answer
128 views

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 ...
6
votes
1answer
96 views

Stabilizers of multilinear forms

Let $\{e_1,\ldots, e_n\}$ be the standard basis of $\mathbb{C}^n$. Consider the $m$-multilinear form $$v=\sum_{i=1}^n e_i^{\otimes m}\in (\mathbb{C}^n)^{\otimes m}$$ and consider the action of $\text{...
4
votes
0answers
61 views

Stabilizer of the Bryant-Harvey associative calibration

View $\mathbb{R}^{4n+4} = \mathbb{H}^{n+1}$ with its standard inner product. Right multiplication $R_I, R_J, R_K$ by the unit quaternions $I,J,K$ defines orthogonal complex structures on $\mathbb{R}^{...
3
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0answers
193 views

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 ...
0
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0answers
42 views

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 $(a1,...ak)$,...
4
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0answers
108 views

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 ...
2
votes
1answer
165 views

Trivial rational solution of a system of hyperplanes

Let us consider a vector space $ V $ over $ \mathbb{Q} $ of dim $6$. We denote all the two dimensional subspace in $ V $ by $ G(2,6) $ (The Grassmanian variety). One can define a map $ p $ from $ G(2,...
0
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0answers
36 views

Range of linear transformation of a natural vector

Given a natural variable vector and its upper bound $\vec{x}=[x_1,x_2,\dotsc, x_n]^\top,\vec{X}=[X_1,X_2,\dotsc,X_n]^\top$, where $x_i,X_i\in\mathbb{N}\wedge x_i<X_i,\ i=1,2,\dotsc,n$. With a ...
3
votes
1answer
262 views

Maximal common isotropic subspace for a finite family of skewforms

Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
3
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0answers
127 views

Polynomial invariant relating the circumradius and sides of a cyclic polygon

This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon. This invariant is discussed briefly in the seminal paper On the Areas ...
2
votes
2answers
135 views

$O(n)$ Polynomial invariant of symmetric tensors

I apologize in advance if this question is not up to the level of research level questions on Math overflow. I am a complete outsider to invariant theory/representation theory and would like someone ...
1
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0answers
179 views

Geometric interpretation for non-simple $k$-blades [closed]

In geometric algebra, a simple k-blade may be defined as one which can be written as the outer product of $k$ vectors. For example, a 2-blade $A$ is one which may be factored as $A=\mathbf{a}\wedge\...
0
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1answer
100 views

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 ...
3
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0answers
122 views

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. ...
2
votes
0answers
194 views

What is the name of this tensor?

A matrix M is usually called a hollow matrix if all of its diagonal elements are zero: $$ M_{pp} = 0, \quad \forall \: p. $$ We can generalize this to an $n$-way tensor T, such that: $$ T_{p_1 \cdots ...
4
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0answers
111 views

Pseudo-tensor- and tensor-densities: Sections of what bundle?

Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle $$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$ ...
3
votes
1answer
256 views

Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$

I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction: First, I take the following partial ...
8
votes
1answer
220 views

Kulkarni-Nomizu square root of the Riemann tensor

Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
7
votes
1answer
495 views

Strategies for bounding the spectral norm of a tensor?

Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by $$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$ (...
1
vote
1answer
117 views

Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$ and let $e_1$, $e_2$, $\ldots$, $e_{10}$ denote its rows. For $i\in \{1,5 \}$, ...
2
votes
1answer
72 views

Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which there exists a matrix $X$ ...
3
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0answers
63 views

Bunch of matrices with vanishing permanents

$\DeclareMathOperator{\Per}{Per}$ $\newcommand{\oI}{{\overline I}}$ $\newcommand{\oJ}{{\overline J}}$ Is it possible to classify pairs $(A,B)$ of square, nonsingular matrices over a field of prime ...
2
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0answers
121 views

Distinguishing $0/1$ unimodular or singular matrices having $\mathsf{Permanent}\in\{0,1\}$?

Let $\mathcal T_n=\{M\in\{0,1\}^{n\times n}:\mathsf{Per}(M)=\mathsf{Det}(M)\wedge\mathsf{Det}(M)\in\{0,1\}\}$ (restricted set unimodular or singular having permanent and determinant identical). $\...
2
votes
2answers
334 views

Can the eigenvalues of a real symmetric tensor be complex?

Let $T$ be a fully symmetric tensor of rank $3$ and size $N$. Using the following definition of eigenvalues, let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that: \begin{equation} \sum_{jk}^...
1
vote
1answer
98 views

Algebraic structure of the space of multiaffine maps

Let $V$ be a vector space over a field $\mathbb F$ and $k$ some natural number. It isn't hard to show that the space of multiaffine maps $V^{[k]}\to\mathbb F$ decomposes as a direct sum of vector ...
4
votes
1answer
151 views

Characterizing zero sets of bilinear maps

Let $V,W$ be vector spaces and $X\subset V\times W.$ If $X$ is the zero set of a collection of bilinear maps then it satisfies the following properties: $(0,w),(v,0)\in X$ for all $v,w.$ If $(v,w)\in ...
3
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0answers
183 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} \...
3
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0answers
72 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 ...
1
vote
0answers
115 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 ...
6
votes
1answer
604 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 $\bigwedge^{p}E$ ...
1
vote
1answer
98 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-...
4
votes
1answer
151 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 $$ \mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n $$ ...
1
vote
0answers
100 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 = ...
3
votes
1answer
153 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) ...
16
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0answers
505 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 ...
0
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0answers
86 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 ...
2
votes
0answers
105 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 $\...
9
votes
1answer
399 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 ...
4
votes
1answer
215 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 \...
1
vote
1answer
53 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,\...
13
votes
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 ...
2
votes
0answers
130 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 ...
2
votes
0answers
58 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\...
1
vote
0answers
42 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}^...
2
votes
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 ...
1
vote
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 ...
2
votes
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 ...
2
votes
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{...
1
vote
0answers
57 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
votes
1answer
318 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 ...