Questions tagged [multilinear-algebra]
Tensors, multivectors, wedge products, multilinear maps, exterior (Grassmann) algebras.
170
questions
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Linear system with matrix as a variable
I have the following two linear systems:
$$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$
$$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$
Both $A,B$ are $2 \times 2$ matrices ...
2
votes
0
answers
112
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Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$
Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
6
votes
2
answers
462
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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 ...
15
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2
answers
1k
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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 $\tilde q$ ...
12
votes
1
answer
885
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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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
2
votes
1
answer
182
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Orthogonal complements in exterior powers
I previously asked this on Mathematics Stack Exchange, to no result:
Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge ...
1
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0
answers
119
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Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
3
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0
answers
468
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Geometric interpretation for non-simple $k$-vectors [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\...
1
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0
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168
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The conditions to determine whether multivector $\Lambda\in\wedge^k V$ is decomposable
In Section 5, Chapter 1 of the famous book "Principles of algebraic geometry" by Griffiths and Harris, there are two equivalent conditions to determine whether a multivector $\Lambda\in\...
8
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3
answers
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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$ ...
3
votes
0
answers
57
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Automatic complete boundedness for bilinear and multilinear maps
$\newcommand{\cb}{\mathrm{cb}}$Let $T : X \rightarrow Y$ be a bounded linear map between Banach spaces. We have the following results concerning automatic complete boundedness:
$\|T : X \rightarrow \...
1
vote
1
answer
204
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Third order matrix differential norm
Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
11
votes
2
answers
1k
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Tensor rank of anti-symmetric tensor
Let $V$ be a vector space of dimension $n$. Let us consider $V^{\otimes n}=V\otimes V \ldots \otimes V$. This vector space contains one dimentional vector space $\wedge^n V$. My question is does it ...
4
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1
answer
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Etymology “Kulkarni–Nomizu product”
$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...
5
votes
1
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358
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Waring rank of monomials, and how it depends on the ground field
The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$:
$$
p = \sum_{...
11
votes
1
answer
342
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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 ...
1
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1
answer
116
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Is it possible to simplify the coefficient matrix for large values of $x$?
If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
4
votes
1
answer
182
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Singular value decomposition for tensor
I am looking at (the limitation of) the extension of the singular value decomposition to tensors. I would like to show that there is a tensor $A_{i,j,k}$ that cannot be decomposed in the following ...
1
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1
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228
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Two unknowns: one vector, one scalar, one equation
I would like to know if this equation is solvable for $a$ and $\alpha$:
\begin{equation}
\Sigma = \Gamma + a \left( \alpha 1^\top + 1\alpha^\top \right) +a^2 b
\end{equation}
$\Sigma$ & $\Gamma$ ...
0
votes
0
answers
67
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On comparing the nuclear norm with the Hilbert-Schmidt norm for symmetric tensors
I am interested in the special case of a symmetric tensor $T_{i_1,\ldots,i_k}$ of rank $k$, where each index, say $i_\kappa$, where $1 \leq \kappa \leq k$, runs from $1$ to $2$. The entries of such a ...
1
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0
answers
83
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Continuous choice of null directions for a family of bilinear forms
Let $E$ and $F$ be (real) Hilbert spaces, where $\dim E = \infty$ and $1 \leq \dim F < \infty$. Let $T : E \to \operatorname{Sym}(F \times F,\mathbb{R})$ be a continuous linear map, where $\...
2
votes
2
answers
258
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Optimizing a multilinear function over the vertices of the cube
Suppose I have $n$ Boolean variables $x_1,\dots,x_n$, and an objective function of the form $f(x_1,\dots,x_n) = \sum_{a_1,\dots,a_n}c_{a_1,\dots,a_n} x_1^{a_1} \cdots x_n^{a_n}$ with $(a_1,\dots,a_n) \...
4
votes
1
answer
1k
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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{...
6
votes
1
answer
568
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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: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
2
votes
0
answers
70
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Nontrivial Invariants of trilinear functionals
The group $\operatorname{SL}(n_1,\mathbb{C}) \times \operatorname{SL}(n_2,\mathbb{C}) \times \operatorname{SL}(n_3,\mathbb{C})$ acts on ${\mathbb C}^{n_1} \otimes {\mathbb C}^{n_2} \otimes {\mathbb C}^...
31
votes
4
answers
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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 ...
0
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0
answers
105
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Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
4
votes
1
answer
183
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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
1
answer
133
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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{...
0
votes
1
answer
122
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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 ...
4
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0
answers
83
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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}^{...
6
votes
2
answers
310
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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 ...
3
votes
0
answers
198
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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
votes
0
answers
52
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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
votes
1
answer
352
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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
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...
4
votes
0
answers
130
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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
1
answer
195
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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,...
3
votes
1
answer
341
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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
votes
0
answers
169
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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
2
answers
368
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$O(n)$ 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 ...
3
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0
answers
137
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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. ...
7
votes
1
answer
468
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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 ...
2
votes
0
answers
199
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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:
$$ T_{p_1 \cdots ...
4
votes
0
answers
215
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Pseudo-tensor- and tensor-densities: Sections of what bundle?
Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
3
votes
1
answer
277
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Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$
I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...
9
votes
1
answer
349
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Kulkarni-Nomizu square root of the Riemann tensor
Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
7
votes
1
answer
944
views
Strategies for bounding the spectral norm of a tensor?
Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by
$$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$
(...
1
vote
1
answer
174
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
1
answer
81
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
votes
0
answers
75
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 ...