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Questions tagged [multilinear-algebra]

For questions about the extension of linear algebra to multilinear transformations of vector spaces.

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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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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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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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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 ...
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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 ...
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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)\...
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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 ...
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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 ...
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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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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, ...
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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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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 ...
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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]=\{...
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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 ...
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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}}&...
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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,...,...
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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 ...
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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 ...
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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 $m\times n$ ...
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1answer
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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 ...
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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 ...
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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 $\mathfrak{F}_{(\mathfrak{U}\times\mathfrak{V})}...
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$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.
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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} \...
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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 $A$ ...
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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 ...
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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 ...
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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 \...
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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((...
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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 ...
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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) = \...
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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 $\...
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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 ...
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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 ...
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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 \...
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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)...
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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 (...
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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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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{...
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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 $d_1$ ...
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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 ...
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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 ...
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$f'$ 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 $\sup\limits_{c\in O}\Vert f'(c) \Vert<\infty;$ $ii$. $...
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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 ...
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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-\{...
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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 ...
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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 ...
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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^* ...
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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. ...
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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 \...