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

**3**

votes

**1**answer

976 views

### Nonlinear matrix equation

Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw=\lambda_1v+\lambda_2w$
$Aww^TAv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \lambda_3$ are ...

**10**

votes

**3**answers

496 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. ...

**2**

votes

**1**answer

134 views

### How to characterize the dual of an isotropic hyperplane?

Hi there! I have a very simple question, which requires an expert in multilinear algebra.
$V$ is an $n$-dimensional vector space, and $\omega\in V^\ast\wedge V^\ast$ is a skew-symmetric form on it. ...

**1**

vote

**0**answers

192 views

### Singular quadratic space

Let $(V,b)$ a symmetric bilinear space. An old theorem of Witt says that if $(V,b)$ is regular, then given a subspace $W$ of $V$ and an isometry $\sigma: W \to V$, there exists an isometry $\Sigma: V \...

**5**

votes

**4**answers

1k views

### Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms

Hello,
Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vectorspace and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$.
The Polarisation ...

**7**

votes

**1**answer

315 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 \...

**4**

votes

**3**answers

3k 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 $\mathbb{R}^n$ ...

**6**

votes

**2**answers

352 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 ...

**4**

votes

**2**answers

215 views

### Reference for Tensors on graded spaces needed

Is there a good introduction to
1.) Tensor (co)algebras on graded vector spaces ?
2.) Tensor (co)algebras on graded modules ?
In the research field of $L_\infty$-algebras there is some stuff, but ...

**3**

votes

**1**answer

783 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 ...

**0**

votes

**0**answers

531 views

### modified bessel fucntion of the third kind

Hi I'm doing a computation where the modified bessel function of the third kind is the main source of computational strain, we are using a 10,000 dimension's for our distribution, is there any easier ...

**-2**

votes

**1**answer

302 views

### A Matrix equation

Let $A$ and $B$ be two $n \times n$ full-rank matrices.
Let $XAY = B$ be the given equation where $X$ and $Y$ are unknown $n \times n$ matrices. We know that $Vec(B) = (Y^{T} \otimes X)Vec(A)$. Under ...

**8**

votes

**1**answer

1k 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 ...

**8**

votes

**1**answer

833 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 ...

**11**

votes

**0**answers

497 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 ...

**4**

votes

**2**answers

499 views

### Tensor and Hom objects for finite flat group schemes

Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to ...

**6**

votes

**1**answer

790 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) \...