Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
0
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1answer
41 views
How to prove that the definition of exterior product of differential forms is not ambiguous?
In page 91 of book A Visual Introduction to Differential Forms and Calculus on Manifolds the exterior product of two differential forms $\alpha \in \bigwedge^{r}(\mathbb{R}^n)$ and $\beta \in \...
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0answers
54 views
How to determine an integral of a differential form?
Let
$$ \eta = x^2 \, dy \wedge dz + yx\,dz \wedge dx + z^3 \, dx \wedge dy $$
Can you show me how to calculate :
$$ \int_{\Phi} \eta, $$
where $ \Phi $ is supposed to be the parametrization of ...
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0answers
5 views
Relationship between anisotropic and negative/positive definite
Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false:
$(V,<->)$ is ...
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0answers
54 views
Determining values of multilinear forms/ differential forms
I started getting into the topic of multilinear forms and differential forms.
I find it quite hard to get into.
if I solve $$ \Phi ( \begin{pmatrix} 1 \\ 2\\3 \end{pmatrix} , \begin{pmatrix} 4 \\5\\6 ...
1
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1answer
14 views
Convert from one tensor canonical form to another
Suppose we have two canonical forms $A, B \in \mathbb{F}_2^{2 \times 2 \times 2}$ of a 3-dimensional tensor product space over the finite field with two elements, where
$A = e_1 \otimes e_2 \otimes ...
1
vote
1answer
35 views
Prove that the image of the bilinear application is not a vector space
Prove that the image of the bilinear application
$$
\mathbb{R}^2\times\mathbb{R}^2
\ni \big((x_1,x_2),(y_1,y_2)\big)
\mapsto
\varphi \big((x_1,x_2),(y_1,y_2)\big) =
(x_1y_1,x_1y_2,x_2y_1,x_2y_2)\...
1
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0answers
19 views
Switch to another object of same tensor 'rank' to introduce a different class of multilinear maps
My idea is very simple: we know that 'tensors' are a class of multilinear maps or object that "maps" in a (multi-)linear manner geometric vectors, scalars.
a multilinear map is a function of ...
0
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1answer
15 views
Multilinear form is alternating if the underlying field has characteristics different from 2
Let $F$ be a field, $V$ be a $F$-vector space of dimension $n$ and $L^k(V,F)$ be the space of $k$-multilinear forms $f:V^k\rightarrow F$.
I was reading Henri Cartan's Differential Form, where he ...
0
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0answers
20 views
Is $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]\rightarrow$ $\mathbb{F}_2$ multilinear?
$t$ : $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]\rightarrow$ $\mathbb{F}_2$
$t(f,g,h):=f(0)g(0)h(1)+f(0)g(1)h(0)+f(1)g(0)h(0)$
How can I know if this is multilinear? Do I have to ...
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0answers
26 views
Inverse function being onto.
Follow up on this question:
As you can see from this link
$\Phi=f^{-1}\circ h$ is well defined and maps $[c,d]$ onto $[a,b]$ now in my textbook the next theorem following what I posted in the link, ...
0
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0answers
32 views
Expressing tensor product in terms of symmetric product and exterior product
Let $V$ be a $k$-vector space. Is there a way to express $T_n(V)$ (n-fold tensor of $V$) in terms of symmetric products $Sym_i(V)$ and exterior products $\Lambda^j(V)$ ?
Maybe this is not a very well-...
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0answers
27 views
The Universal Mapping Property of a free vector space.
Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:X\to V$ satisfying the following universal mapping property.
I ...
4
votes
5answers
79 views
If there are $2$ linearly independent vectors $x,y \in X$ such that $||x+y||=||x||+||y||$, then the unit sphere $S(X)$ contains an interval
Let $S(X)= \{x \in X: ||x||=1\}$ be the unit sphere in $X$. Assume that there are $x,y\in X$ linearly independent such that $||x+y||=||x||+||y||$. Prove that $S(X)$ contains the following set:$[x,y]=\{...
1
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0answers
63 views
Are antilinear forms part of the tensor algebra of a $\mathbb{C}$-vector field?
Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $\mathcal{T}(V)$ be its tensor algebra. If $K$ is either $\mathbb{R}$ or $\mathbb{C}$, then every multilinear ...
1
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0answers
51 views
How to find the value of this determinant?
I'm wondering how to find the value of this determinant.
$$\left[ {\begin{array}{*{20}{c}}
0&{{x_1}}&{{x_2}}&{{x_3}}& \ddots &{{x_n}} \\
{{x_1}}&0&{{x_1}}&{{x_2}}&...
0
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0answers
19 views
multilinear rank of a tensor
let $A$ be a tensor in $R^{n_1\times...\times n_d}$
if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,...
4
votes
1answer
66 views
Is the length of the multilinear cross product given by this formula? If not, what is the correct formula?
This is an attempt to make this question more specific.
We can compute the length of the cross product in $\mathbb{R}^3$ using the formula:
$$|x\times y| = \sqrt{|x|^2|y|^2 - |x\cdot y|^2}$$
This ...
2
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0answers
51 views
Musical Isomorphisms
I'm studying from Fecko's Differential Geometry and Lie Groups for Physicists, and in the part introducing metric tensors, Fecko introduces the musical isomorphisms between the tangent and cotangent ...
0
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0answers
27 views
Tensors, Co- and Contravariance
Recently I have started to learn about Tensor-Calculus and in my head I made some connections/conclusions which I want to know whether I'm right or wrong about.
As an example consider a $m\times n$ ...
2
votes
1answer
61 views
Lagrange identity for determinants
Let A $\in M_{(n-1)},n(\mathbb{R})$ and for each $1\leq j \leq n$,let $A_j$ the matrix obtained from A by removing the j-th column. Show that:
$det (AA^t)= \sum\limits_{j=1}^n det(A_j)^2$
My first ...
1
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0answers
47 views
finding a projection on tensor spaces
I want to specify a projection from $V \otimes W$ to itself . How should I go about doing this ? And, can I charcterize all such projections that are possible ?
Edit 1 : $V$ and $W$ are any two ...
1
vote
2answers
63 views
Question about the role of Quotient Spaces as Tensor Producs
According to Steven Roman's Advanced Linear Algebra, the very first objects that is required to construct the Tensor Product lies on a free vector space $\mathfrak{F}_{(\mathfrak{U}\times\mathfrak{V})}...
-1
votes
1answer
34 views
$u,v,w$ are distinct vectors show the following is also a basis [closed]
Let $u,v,w$ be distinct vectors of vector space over $\mathbb C$, such that $\{u,v,w\}$ is a basis of $V$.
Show that $\{u−(1+i)v, u+v+w, −2iu\}$ is also a basis.
1
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0answers
34 views
Silly doubt about a particular tensor product
In physics we have a particular tensor called inertia tensor. In abstract notation, this tensor can be written as:
$$ \textbf{I} = m [\langle \textbf{r},\textbf{r} \rangle \textbf{Id} - \textbf{r} \...
1
vote
1answer
36 views
Continuity of symmetric multilinear maps
The following is Exercise 2.11-2 from Linear and Nonlinear Functional Analysis with Applications by Philippe G. Ciarlet.
Let $X$ and $Y$ be two normed vector spaces over the same field, and let $A$ ...
0
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0answers
38 views
Equivalence between dual vectors and dual covectors
So I'm studying the treatments of vectors and covectors across multiple resources and this is an area where different authors take wildly different approaches; but all of them seem to lead to the same ...
2
votes
1answer
87 views
Basis vectors and Basis Covectors
If I define a vector so that:
$$ \vec v = v^i e_i = v_i e^i $$
Where $ e_i e^j = \delta _i^j$, then its fair to say that $e_i$ and $e^i$ are dual vectors to eachother. Now from this, we can show ...
2
votes
1answer
71 views
Higher total derivatives (Frechét derivatives)
This issue I just cannnot resolve, so I'd highly appreciate your help.
Let $a_1, ... , a_n \in \mathbb{R}^k$ with $k$ a natural positive number. If we consider the function $$ W: \mathbb{R}^k \to \...
0
votes
1answer
48 views
Proof verification : T-invariant subspace has T-invariant complement
I was asked to prove the following Lemma :
Lemma: Let $\varphi$ be a symmetric or alternating bilinear form , V a finite-dimensional space , T an isomorphism $T:V\rightarrow V$ such that $\varphi((...
0
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0answers
54 views
How to get the linear combination of a k-form?
Let M be a smooth manifold then a k-form on it is defined as follows:
$w: M \rightarrow \Lambda^k T_p M \\ p \mapsto w_p$
Where $w_p$ is a k-form on the tangent space. We call such an element ...
0
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0answers
14 views
Determinant from a Rank r perturbation from Golub paper
In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...
0
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0answers
21 views
About the symmetric multilinear maps
Consider $E$ and $F$ two vector spaces over $\mathbb{R}$, and $f : E^n \longmapsto F$ a $n$-linear map.
Assume that $f$ is symmetric, i.e. for all $(v_1,...,v_n) \in E^n$, for all permutation $\...
0
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1answer
30 views
A natural identification of the second exterior power of linear operators ?
Let $K$ be a field of characteristic zero and $V=K^n$. Let $A\in M(n,K)$, so we can think of $A$ as a linear map $A: V \to V$ be a linear map . Let $\wedge^2 V$ be the second exterior power of $V$ and ...
0
votes
2answers
88 views
Iterated integrals as pre-sheaves
Let $n \in \{ 1, 2, 3, \ldots \}$ be fixed and set $N = \{ 1, \ldots, n \}$.
Let $X_1, \ldots, X_n$ be measure spaces and for $I = \{ i_1, \ldots, i_m \} \subseteq N$ set $X^I = X_{i_1} \times \cdots ...
1
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0answers
25 views
Kronecker product SVD Error bound
The Kronecker Product SVD (KPSVD) is defined here. Given a target rank $r$, what is the error bound in terms of singular values $\sigma_i$ for $\|A - A_r\|_F$, where $A_r = \sum_{i=1}^r \sigma_iU_i \...
0
votes
1answer
53 views
Endomorphism between direct sum of modules $M := M_1 \oplus M_2$
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\End{End}$Let $M_1$, $M_2$ be a $R$-modules and $M := M_1 \oplus M_2$ if $ \pi_1$, $\pi_2$ are the respectives projections and any $ \phi \in \End_R(M)...
7
votes
1answer
225 views
When is the set of tensors of rank at most $r$ closed?
Consider the vector space $V=A_1\otimes A_2\otimes\dots\otimes A_k$ where $k\geq 2$ and let $\sigma_r$ be the set of tensors in $V$ of rank at most $r$. It seems clear that $\sigma_r$ is Zariski (...
1
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1answer
66 views
Determinant of a special type of skew symmetric matrix with complex entries
Let $a_1,...,a_{2n} \in \mathbb C$ and $A=[b_{ij}]\in M(2n,\mathbb C)$ such that $A^T=-A$ and $b_{ij}=a_ia_j,\forall i<j$.
Can we find a nice expression for determinant of $A$?
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votes
1answer
28 views
on the matrix representation of a canonical linear map on the vector space of $4\times 4$ skew-symmetric matrices
For $t_1,t_2,...,t_6\in \mathbb R$, let $P_{(t_1,t_2,...,t_6)}=\begin{pmatrix} 0&t_1&t_2&t_3\\ -t_1&0&t_4&t_5\\ -t_2&-t_4 &0&t_6\\-t_3&-t_5&-t_6&0\end{...
1
vote
1answer
37 views
Does the differential of an augmented dga algebra fix the augmentation ideal?
I am reading about the bar/cobar construction in the book Algebraic Operads. The differential on the bar construction of a augmented dga algebra $A$ is a sum of two differentials $d_1+d_2$ where $d_1$ ...
0
votes
1answer
31 views
Doubt on the construction of canonical exemple of the existence of Tensor Product Vector Space
My text is based on these notes:
http://outcomes.enquiringminds.org/definition-and-construction-of-the-tensor-product/
http://www-users.math.umn.edu/~broom010/doc/TensorProduct.pdf
There are one ...
1
vote
0answers
85 views
Value changes after using a permutation in a wedge-product
We defined the wedge-product as follows: $ w \wedge \mu = \frac{(k + l)!}{k!l!} Alt (w \otimes \mu) $ (here you'll find additional information).
In the context of a proof we came to a point where we ...
0
votes
1answer
65 views
$f'$ is bounded and $\sup\limits_{c\in O}\Vert f'(c) \Vert<\infty$ if and only if $f$ is globally Lipschitz
Let $f:O\subset \Bbb{R}^n\to\Bbb{R}^m$ be differentiable and O be convex. Prove that the following are equivalent:
$i$. $f'$ is bounded and $\sup\limits_{c\in O}\Vert f'(c) \Vert<\infty;$
$ii$. $...
2
votes
0answers
67 views
Prove using Implicit Function Theorem, that the equation $u^3-2u^2+uv+\operatorname{Id}=0$ has a solution for $|v|$ small enough
I want to prove that the equation \begin{align}u^3-2u^2+uv+\operatorname{Id}=0\end{align} has a solution for $|v|$ small enough. My goal is to use Implicit Function Theorem. I assume that the ...
1
vote
2answers
39 views
Prove that $f$ and $g$ are differentiable on $\Bbb{R}^n$ and $\Bbb{R}^n-\{0\}$, respectively and compute $f'(x)$ and $g'(x)$
Let $f,g:\Bbb{R}^n\to\Bbb{R} $ be defined by $f(x)=\langle x,x\rangle$ and $g(x)=\sqrt{\langle x,x\rangle}\,,$ respectively.
Prove that $f$ and $g$ are differentiable on $\Bbb{R}^n$ and $\Bbb{R}^n-\{...
0
votes
3answers
37 views
Can a Rank Three Tensor act as a Trilinear, Bilinear, or Linear Map?
Can a rank three tensor act as a trilinear, bilinear, and linear map? Similarly, a matrix (a representation of a rank two tensor) can be bilinear, taking in two vectors and spitting out a scalar for ...
2
votes
1answer
82 views
Alternating multilinear function - understanding of specific proof
I was trying to understand a proof that the Plücker coordinates satisfy the Plücker relations. However, I assume that my problems come from a misunderstanding/lack of knowledge about multilinear ...
1
vote
0answers
43 views
Objects of mixed tensor spaces
So say I have a real vector space $V$ of dimension $n$. Consider the tensor space $T^{(2, 0)}(V) = V \otimes V$, note that $V \otimes V$ is isomorphic to the vector space of multilinear maps from $V^* ...
2
votes
2answers
78 views
Contraction of Tensors is Independent of the Choice of Basis
Definition: Let $ T:(V^*)^k \times V^l \rightarrow \mathbb{R}$ be a tensor of type $ (k,l)$. Let $ \{v_1,...,v_n\}$ be a basis of $V$ and $ \{v^{1^*},...,v^{n^*}\}$ be the corresponding dual basis. ...
2
votes
1answer
47 views
A form $\omega$, of degree $2$ in $U$ such that $\omega \wedge \alpha = \omega \wedge \beta = 0$, prove that $\omega = f\cdot \alpha \wedge \beta$
Let $\alpha, \beta$ $1$-forms in the open $U \subset \mathbb{R}^{3}$ such that $\alpha \wedge \beta \neq 0$ in every $x \in U$. If a form $\omega$, of degree $2$ in $U$ is such that $\omega \wedge \...
