Questions tagged [multilinear-algebra]
Tensors, multivectors, wedge products, multilinear maps, exterior (Grassmann) algebras.
170
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2
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0
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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 $\mathcal{B}\subset \mathbb{R}^{k}$ (...
12
votes
1
answer
884
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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
180
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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 $$\langle u_1\wedge \cdots \wedge ...
15
votes
2
answers
1k
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Positive quadratic polynomial
Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...
1
vote
1
answer
199
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Linear system with matrix as a variable
I have the following two linear systems:
$$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$
$$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$
Both $A,B$ are $2 \times 2$ matrices ...
1
vote
0
answers
118
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Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
1
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0
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168
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The conditions to determine whether multivector $\Lambda\in\wedge^k V$ is decomposable
In Section 5, Chapter 1 of the famous book "Principles of algebraic geometry" by Griffiths and Harris, there are two equivalent conditions to determine whether a multivector $\Lambda\in\...
3
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0
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57
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Automatic complete boundedness for bilinear and multilinear maps
$\newcommand{\cb}{\mathrm{cb}}$Let $T : X \rightarrow Y$ be a bounded linear map between Banach spaces. We have the following results concerning automatic complete boundedness:
$\|T : X \rightarrow \...
1
vote
1
answer
198
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Third order matrix differential norm
Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
4
votes
1
answer
362
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Etymology “Kulkarni–Nomizu product”
$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...
5
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1
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354
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Waring rank of monomials, and how it depends on the ground field
The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$:
$$
p = \sum_{...
11
votes
1
answer
340
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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
vote
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
vote
1
answer
226
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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
66
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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
255
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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) \...
2
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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}^...
0
votes
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
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132
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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{...
4
votes
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}^{...
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
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0
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129
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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
339
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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
367
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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
votes
0
answers
461
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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\...
0
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1
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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 ...
3
votes
0
answers
136
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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. ...
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}.$$
...
3
votes
1
answer
276
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Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$
I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...
9
votes
1
answer
347
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Kulkarni-Nomizu square root of the Riemann tensor
Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
7
votes
1
answer
938
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Strategies for bounding the spectral norm of a tensor?
Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by
$$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$
(...
1
vote
1
answer
174
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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
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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
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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
votes
0
answers
137
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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
2
answers
505
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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
1
answer
138
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
1
answer
249
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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
votes
0
answers
251
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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
votes
0
answers
107
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
0
answers
136
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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
1
answer
1k
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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
1
answer
165
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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-...