Questions tagged [multilinear-algebra]
Tensors, multivectors, wedge products, multilinear maps, exterior (Grassmann) algebras.
170
questions
12
votes
1
answer
849
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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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
- 42.2k
2
votes
1
answer
159
views
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 $$\langle u_1\wedge \cdots \wedge ...
- 6,156
15
votes
2
answers
1k
views
Positive quadratic polynomial
Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...
- 42.2k
0
votes
1
answer
177
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Linear system with matrix as a variable
I have the following two linear systems:
$$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$
$$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$
Both $A,B$ are $2 \times 2$ matrices ...
- 109
1
vote
0
answers
107
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Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
- 461
1
vote
0
answers
159
views
The conditions to determine whether multivector $\Lambda\in\wedge^k V$ is decomposable
In Section 5, Chapter 1 of the famous book "Principles of algebraic geometry" by Griffiths and Harris, there are two equivalent conditions to determine whether a multivector $\Lambda\in\...
- 41
3
votes
0
answers
55
views
Automatic complete boundedness for bilinear and multilinear maps
$\newcommand{\cb}{\mathrm{cb}}$Let $T : X \rightarrow Y$ be a bounded linear map between Banach spaces. We have the following results concerning automatic complete boundedness:
$\|T : X \rightarrow \...
- 493
1
vote
1
answer
146
views
Third order matrix differential norm
Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
- 39
4
votes
1
answer
315
views
Etymology “Kulkarni–Nomizu product”
$\newcommand\KN{\mathbin{\bigcirc\mspace{-20mu}\wedge\mspace{3mu}}}$In the context of (pseudo)-Riemmian geometry, the Kulkarni–Nomizu product is defined to be an operation $\KN$, which takes two ...
- 1,061
0
votes
0
answers
46
views
Taylor series for multidimensional orthant probability
Here we have a solution as Measuring Solid Angles Beyond Dimension Three, https://doi.org/10.1007%2Fs00454-006-1253-4. As we know there is no closed-form expression for the probability of higher than ...
- 25
5
votes
1
answer
317
views
Waring rank of monomials, and how it depends on the ground field
The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$:
$$
p = \sum_{...
- 5,513
11
votes
1
answer
323
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 ...
- 291
1
vote
1
answer
113
views
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 ...
- 111
4
votes
1
answer
175
views
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 ...
- 2,169
1
vote
1
answer
221
views
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}
$\Sigma$ & $\Gamma$ ...
- 13
0
votes
0
answers
53
views
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 ...
- 4,697
1
vote
0
answers
82
views
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 $\...
- 2,073
2
votes
2
answers
234
views
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) \...
- 19k
2
votes
0
answers
70
views
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}^...
- 101
0
votes
0
answers
102
views
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 ...
- 913
4
votes
1
answer
182
views
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 ...
- 913
6
votes
1
answer
130
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{...
- 4,969
4
votes
0
answers
81
views
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}^{...
- 861
3
votes
0
answers
198
views
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 ...
- 758
0
votes
0
answers
51
views
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
0
answers
127
views
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 ...
- 1,423
2
votes
1
answer
192
views
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,...
- 913
3
votes
1
answer
328
views
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 $\alpha$...
- 913
3
votes
0
answers
167
views
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 ...
- 149
2
votes
2
answers
297
views
$O(n)$ 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 ...
- 1,025
3
votes
0
answers
399
views
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\...
- 47
0
votes
1
answer
117
views
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 ...
- 195
3
votes
0
answers
133
views
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. ...
- 31
2
votes
0
answers
199
views
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:
$$ T_{p_1 \cdots ...
- 47
4
votes
0
answers
197
views
Pseudo-tensor- and tensor-densities: Sections of what bundle?
Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
- 811
3
votes
1
answer
274
views
Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$
I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...
- 301
9
votes
1
answer
335
views
Kulkarni-Nomizu square root of the Riemann tensor
Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
- 1,279
7
votes
1
answer
864
views
Strategies for bounding the spectral norm of a tensor?
Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its spectral norm $|A|$ by
$$|A| = \sup_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|_2^k}.$$
(...
- 19k
1
vote
1
answer
157
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 $i\in \{1,5 \}$, ...
user83947
2
votes
1
answer
79
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 $X$ ...
user83947
3
votes
0
answers
73
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 ...
- 22.6k
2
votes
0
answers
136
views
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).
$\...
- 13.5k
2
votes
2
answers
488
views
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}^...
- 87
1
vote
1
answer
131
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 ...
- 225
4
votes
1
answer
227
views
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 ...
- 225
3
votes
0
answers
240
views
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} \...
- 139
3
votes
0
answers
101
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 ...
- 17.2k
1
vote
0
answers
134
views
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 ...
- 326
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 $\bigwedge^{p}E$ ...
- 1,129
1
vote
1
answer
161
views
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-...
- 47