Questions tagged [multilinear-algebra]

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

Filter by
Sorted by
Tagged with
35 votes
4 answers
2k views

Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix $$ \left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ...
Giovanni Moreno's user avatar
31 votes
4 answers
2k 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 ...
fosco's user avatar
  • 13k
26 votes
3 answers
3k 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}_{...
Arthur's user avatar
  • 1,379
20 votes
3 answers
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 ...
Nik Weaver's user avatar
19 votes
1 answer
2k views

Are the following identities well known?

$$ x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right) $$ $$ \begin{eqnarray} x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\ &-...
Hideaki Okazaki's user avatar
17 votes
1 answer
759 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 ...
Francois Ziegler's user avatar
17 votes
0 answers
586 views

Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
Bugs Bunny's user avatar
  • 12.1k
16 votes
0 answers
552 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 ...
pregunton's user avatar
  • 976
15 votes
2 answers
1k views

Positive quadratic polynomial

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$. Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$. Is it possible to find a polynomial $\tilde q$ ...
Anton Petrunin's user avatar
14 votes
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 ...
Hailong Dao's user avatar
  • 30.3k
14 votes
2 answers
2k views

Norm of $n$-linear symmetric forms

Let $B$ be a symmetric bilinear form over a Euclidean space $E$. Say that $|B(v,v)|\le c\|v\|^2$ for every $v\in E$, for some $c\ge0$. Then $$4B(v,w)=B(v+w)+B(v-w)$$ yields $2|B(v,w)|\le c(\|v\|^2+\|w\...
Denis Serre's user avatar
  • 51.5k
13 votes
1 answer
1k views

Co-ends as a trace operation on profunctors

The n-lab site on profunctors (http://ncatlab.org/nlab/show/profunctor) describes profunctor composition as using a co-end to "trace out" the connecting variable: $F\circ G := \int^{d\in D} F(-, d) \...
Aleks Kissinger's user avatar
12 votes
1 answer
885 views

Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$. Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
Anton Petrunin's user avatar
12 votes
2 answers
2k views

basics of classification of trilinear forms (when is it non-discrete)

Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,..,n_1$, $j=1,..,n_2$, $k=1,..n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices. What is ...
Dmitry Kerner's user avatar
12 votes
3 answers
595 views

Example of a form linear in infinitely many variables ?

We all know plenty of examples of multilinear forms in finitely many variables (e.g. determinants). However, I am missing an interesting example of a form in infinitely many variables, linear in each. ...
Matthieu Romagny's user avatar
12 votes
0 answers
597 views

On a tentative generalization of the Schmidt decomposition

Background I am a PhD student in Physics and I am currently developing quite refined computer codes that allow to simulate many-body quantum systems living on a lattice. The difficulty resides in ...
user14548's user avatar
  • 141
11 votes
2 answers
1k 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 ...
Klim Efremenko's user avatar
11 votes
1 answer
342 views

Why are these graphs coming from 9-dimensional alternating trilinear forms so symmetric?

Let $\phi(x,y,z)$ be an alternating trilinear form on a space $V$ over a field $K$. Let $u \in \mathbb{P}(V)$ be a projective point over $V$, then we say that the rank of $u$ is equal to the rank of ...
Ward Beullens's user avatar
11 votes
1 answer
539 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 ...
Steven Sam's user avatar
10 votes
5 answers
3k 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 ...
Felix Wutschke's user avatar
10 votes
3 answers
484 views

Axiomatizing orientation in the complex plane

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the ...
James Propp's user avatar
  • 19.4k
9 votes
1 answer
475 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 ...
A. Gupta's user avatar
  • 336
9 votes
1 answer
2k views

Multilinear generalization of Cauchy-Schwarz inequality

Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties: $(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the ...
Nate Eldredge's user avatar
9 votes
1 answer
545 views

What is the total polarization of the determinant?

Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the total polarization $\...
Giovanni Moreno's user avatar
9 votes
1 answer
349 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 ...
Carlo Mantegazza's user avatar
9 votes
1 answer
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 ...
Ed Segal's user avatar
  • 460
8 votes
3 answers
4k views

n-dimensional "cross product" reference request

I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to. Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ ...
Bill Cook's user avatar
  • 1,177
8 votes
3 answers
611 views

Is there a rank for higher degree homogeneous forms analogous to that of quadratic forms?

Given a quadratic form $Q(x_1, ..., x_n)$, there is a natural notion of rank defined by looking at the rank of the unique symmetric matrix associated to the quadratic form, i.e. we consider the ...
Johnny T.'s user avatar
  • 3,547
8 votes
1 answer
299 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 ...
Graham Denham's user avatar
8 votes
2 answers
770 views

generalizations of Vandermonde matrix to high dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix the maps $$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto (1,x,x^2,\cdots,x^{n-1})...
Quan's user avatar
  • 519
7 votes
1 answer
329 views

Equivalence of exterior forms

Let us start with the following definition. Let $1\leqslant k\leqslant n$ and let $\omega_1,\omega_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega_1$, $\omega_2$ are equivalent, if there exists $T\...
Tatin's user avatar
  • 895
7 votes
1 answer
345 views

Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?

In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...
Drew's user avatar
  • 1,469
7 votes
1 answer
2k views

Spectral sequence of symmetric or exterior algebras?

This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads: Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence ...
Geoffrey's user avatar
  • 727
7 votes
1 answer
228 views

Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$). Can we prove that the following problems are equivalent: $$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
jayki's user avatar
  • 135
7 votes
1 answer
944 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 $$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$ (...
H A Helfgott's user avatar
  • 19.3k
7 votes
1 answer
289 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 $\...
Asaf Shachar's user avatar
  • 6,611
7 votes
1 answer
468 views

Subgroups of the tensor product $A\otimes A$

I have this problem about subgroups of the tensor product of an abelian group $A$ with itself which arises from a complete different setting. I fell into this question studying quandles and quandle ...
marcos's user avatar
  • 457
7 votes
1 answer
353 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 ...
Kevin's user avatar
  • 530
7 votes
1 answer
459 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 ...
Phil-W's user avatar
  • 975
7 votes
1 answer
330 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 ...
user650261's user avatar
7 votes
1 answer
218 views

Some intuition on the $SL_n$-module $V_{[1,1,...,1]}$

(This question highly overlaps with this and also this.) The irreducible ${\sf SL}_{n-1}$-module $V_{[1,1,\ldots,1]}$ is the one providing the minimal projective embedding $\mathbb{P}(V_{[1,1,\ldots,...
Giovanni Moreno's user avatar
7 votes
1 answer
515 views

Question about decomposition of exterior product

In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that $$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong \...
Jesko Hüttenhain's user avatar
6 votes
1 answer
1k 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 $\bigwedge^{p}E$ ...
MathMath's user avatar
  • 1,255
6 votes
1 answer
874 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 ...
SMD's user avatar
  • 480
6 votes
2 answers
734 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 ...
Plethy's user avatar
  • 61
6 votes
2 answers
310 views

Is the triple product in a Freudenthal triple system fully symmetric?

I'm trying to learn about Freudenthal triple systems. Here is the definition given by Helenius [1], start of Section 5: A Freudenthal triple system is a finite-dimensional vector space $V$ over a ...
Vincent's user avatar
  • 2,437
6 votes
1 answer
133 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{...
Ehud Meir's user avatar
  • 4,969
6 votes
1 answer
536 views

Augmentation ideal is finitely generated if and only if $A$ is finitely generated as a $k$-algebra?

Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace $A = \oplus_{i \ge 0} A_i$. One can write $A = A_0 \oplus A_{> 0}$ where we have $A_0 = k^0[x_1, \dots, x_n] = k$ ...
user avatar
6 votes
2 answers
462 views

Alternating multilinear invariants of GL(n) on End (k^n)

Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...
darij grinberg's user avatar
6 votes
1 answer
568 views

Tensor matricizations and their decompositions

Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
qbit-'s user avatar
  • 71