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

**3**

votes

**1**answer

125 views

### A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let $f_{...

**4**

votes

**1**answer

68 views

### Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map
\begin{align}
\alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\
a_1\...

**3**

votes

**1**answer

121 views

### How to embed $S^2\mathbb{C}^2$ into $S^2S^3\mathbb{C}^2$ and get the ideal of the twisted cubic?

Let $X:=x^3$, $Y:=x^2y$, $Z:=xy^2$ and $W:=y^3$ be the 4 independent generators of $S^3\mathbb{C}^2$, and observe that the kernel of the natural epimorphism (total symmetrisation)
$$
p:S^2S^3\mathbb{C}...

**6**

votes

**1**answer

137 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, ...

**5**

votes

**0**answers

79 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: $t_{ijkl} \rightarrow M_{ij, kl}$. ...

**6**

votes

**3**answers

421 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 ...

**1**

vote

**1**answer

141 views

### Is there an algebraic way to characterise the ordinary integral flags?

Fix a vector space $V$ and an integer $1\leq n<\dim V$.
If $\mathcal{I}\subseteq\Lambda^\bullet V^*$ is an ideal, I denote by $\mathcal{I}^i:=\mathcal{I}\cap\Lambda^iV^*$ its $i^\textrm{th}$ ...

**1**

vote

**1**answer

97 views

### A bound on the number of bilinear functions needed in order to obtain the minmax

For $n\in\mathbb N$, let $\Delta(n)=\{x\in\mathbb R^n:x_i\geq 0, \sum_ix_i=1\}$ be the set of probability vectors in $\mathbb R^n$.
Is there a function $m:\mathbb N\to\mathbb N$ such that for any ...

**0**

votes

**1**answer

120 views

### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.
What are some of the standard rational functions that ...

**5**

votes

**1**answer

150 views

### Is the (super-)symmetric power of the exterior algebra free?

Let $V$ be a vector space over $k$ of dimension $m$. (I'm only interested in the case $k=\mathbb{Q}$.) Let $R:=\Lambda^*V$ be the exterior algebra. It carries the structure of a supercommutative ring: ...

**8**

votes

**2**answers

198 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-...

**17**

votes

**1**answer

1k 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 \\
&-...

**4**

votes

**0**answers

103 views

### Cannot multivectors be classified more easily than general tensors?

This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial ...

**8**

votes

**0**answers

176 views

### When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...

**4**

votes

**1**answer

71 views

### Relationship between $N_{A/K}$ and the reduced norm $\text{nr}_{A/k}$?

If $A$ is a finite-dimensional algebra over a field $k$, the usual norm $N_{A/k}: A \to k$ maps $a$ to the determinant of the $k$-linear endomorphism of $A$ given by $x \mapsto ax$. For $A \in \textbf{...

**9**

votes

**1**answer

244 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 $A_0 = k^0[x_1, \dots, x_n] = k$ ...

**30**

votes

**4**answers

1k 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} & ...

**1**

vote

**0**answers

120 views

### Metric(s) on Grassmann Manifold and Plucker Embedding

I'm working on a numerical optimization problem that naturally lives on the Grassmann Manifold Gr$_N(\mathbb{C^M})$, however the objective function is defined on the alternating algebra given by the ...

**5**

votes

**2**answers

220 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\...

**3**

votes

**1**answer

142 views

### Classification of 3-forms in dimension 7

I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant On the geometry of almost ...

**11**

votes

**0**answers

286 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. ...

**2**

votes

**0**answers

44 views

### A tensor equation related to an invariant of a diffeomorphism

Let $M$ be an $n$-dimensional differentiable manifold, $f : U
\rightarrow V$ a diffeomorphism between open neighbourhoods $U$, $V$
of $M$ with $f(x)=x$ for some $x \in U$, and let $R$, $S$, $T$ be
...

**2**

votes

**0**answers

127 views

### Real-rooted polynomials and higher rank matrices

For $A$ and $B$ being matrices of the same dimension and $B$ being rank $1$, one knows that $det(A+tB)$ is a linear polynomial in $t \in \mathbb{R}$. Hence by Taylor series it follows that $det(A + tB)...

**1**

vote

**1**answer

106 views

### Lifting quadratic forms on the cotangent bundle to higher level forms

Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain $\...

**1**

vote

**1**answer

67 views

### Image of skew-symmetric bilinear map which is never zero on linearly independent vectors

I hope this is not too elementary.
Let $B: V\ \times V \to W$ be a skew-symmetric bilinear map where $V$, $W$ are
finite dimensional real vector spaces. Assume that $B (u, v)$ is never zero
for ...

**4**

votes

**1**answer

260 views

### Can the method of small moments prove a bound on the norms of random trilinear forms?

If $F(v_1,\dots,v_k)$ is a $k$-linear form on $\mathbb R^n$, the norm I want to consider is
$$ ||F|| = \sup \frac{ F(v_1,\dots, v_k)}{\prod_{i=1}^k \left|\left|v_i\right|\right|} $$
where the vector ...

**6**

votes

**1**answer

286 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\...

**8**

votes

**1**answer

349 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 $\...

**1**

vote

**1**answer

390 views

### A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way.
...

**5**

votes

**1**answer

186 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 ...

**6**

votes

**3**answers

244 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 ...

**3**

votes

**2**answers

567 views

### Relationship between curvature tensor, algebraic Bianchi identity and sectional curvature

I am currently trying to understand the algebraic Bianchi identity, and I am clearly missing some purely algebraic fact.
Let $M$ be a Riemannian manifold, $R$ its curvature tensor (with index lowered,...

**0**

votes

**3**answers

2k views

### Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$

Let $F$ be an infinite field and let $f \in F[x_{11},x_{12},...,x_{nn}]$ be an arbitrary polynomial in $n^2$ variables. Consider the function $\phi : M_n(F)\longrightarrow F$ defined by $\phi((a_{...

**1**

vote

**1**answer

166 views

### Eigenvalue problem with quadratic constraints

$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = (x_{1},x_{2},.....

**1**

vote

**1**answer

124 views

### Number of linear independent equations

Is there any general rule to find the number of linearly independent equations such that
$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$
where $L_i$ is a ...

**0**

votes

**0**answers

155 views

### When a hyperplane of symmetric forms is determined by a quadric hypersurface?

Let $L$ be a 2D real vector space, $L^*$ its dual, and $\{V,\omega\}$ the symplectic space with $V=L\oplus L^*$ and $\omega$ unambiguously defined by $\omega(l,\lambda):=\lambda(l)$, for all $l\in L$ ...

**3**

votes

**3**answers

727 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 & \...

**1**

vote

**0**answers

575 views

### Tensors as multilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely
$$
V\otimes W := L_2(V^* \times W^*,\Bbb F)
$$
I am also aware that this space is isomorphic to the ...

**5**

votes

**0**answers

152 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 ...

**-1**

votes

**1**answer

241 views

### para-complex structure [closed]

Definition :Let $V$ be a finite dimensional real vector space. A para-complex structure
on $V$ is an endomorphism $K$ :
$V \to V$ such that:
$K$ is an involution, that is $K^2 = Id_V$ ;
The ...

**3**

votes

**2**answers

303 views

### An expression with an alternating trilinear form, written in terms of the determinant and a symmetric bilinear form

I am trying to understand a line from MacLachlan/Reid's The Arithmetic of Hyperbolic 3-Manifolds (it's in 3.4 if you have the book), that seems it should be elementary but I can't seem to find where ...

**3**

votes

**3**answers

755 views

### A basis of the symmetric power consisting of powers

I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:
Let $V$ be a complex vector space ...

**0**

votes

**1**answer

166 views

### Is there a wedge which operates on multiple vector spaces?

Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if ...

**7**

votes

**2**answers

662 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 ...

**1**

vote

**0**answers

154 views

### Non-negative Quadratic forms with Exterior Forms

Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...

**5**

votes

**0**answers

385 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 ...

**0**

votes

**1**answer

302 views

### Inequalities Involving Wedge Product (Reference Request)

Hello,
I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The ...

**3**

votes

**1**answer

198 views

### Decomposability of exterior two-forms

Hello,
The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is welcome....

**0**

votes

**1**answer

131 views

### Nonlinear matrix equation 2

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

**3**

votes

**1**answer

962 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 ...