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

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4
votes
1answer
449 views

Tensor basis change

I have a question regarding tensors and basis changes, however upon searching the web I've found an infinity of definitions for tensors so I'll have to give the one I know first: Given a vector space ...
1
vote
1answer
714 views

Orthogonal Complements property

I have a question about how to prove a certain property of orthogonal complements of vector subspaces. Given $\mathrm E$, a vector space over a commutative field k, define: $$\phi\text{ : } \mathrm E ...
0
votes
0answers
115 views

Second derivative dot product

I was wondering whether I am correct that the second derivative of the dot product is this: Let $f:X \times X \rightarrow \mathbb{K}$ $(x_1,x_2) \mapsto \langle x_1,x_2 \rangle$ Then we have ...
0
votes
1answer
107 views

Finding the determinant of an $n\times n$ matrix… and the inverse

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. ...
4
votes
1answer
103 views

Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...
2
votes
1answer
32 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
1
vote
1answer
46 views

Dimension of (p,q) forms

Let $E$ be a complex vector space of dimension $n$. What is the dimension of the multilinear alternate forms on $E$ of type $(p,q)$ ? I'm sure this is classical but I couldn't find a reference, and ...
2
votes
2answers
386 views

Einstein Notation for product of stacked matrices

Background Information: I recently started using the Einstein summation notation to express certain operations over an "image" $\mathbf{A}$ where to each pixel a square matrix is attached. That is, ...
0
votes
0answers
55 views

Tensor power of a quotient space

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes ...
4
votes
1answer
116 views

Is invariance of a multi-linear form required for co/contra variance?

I'm reading the book: The Absolute Differential Calculus by Levi-Civita to get an idea of the history behind the development of tensor calculus. On page 71 he states: An m-fold covariant is an ...
3
votes
1answer
78 views

Tensor Product and Direct Sum

Let $R$ be a commutative ring with identity and let $\{M_\alpha\}$ be a family of $R$-modules and $N$ another $R$-module. I've tried to show that $$\left(\bigoplus_\alpha M_\alpha\right)\otimes ...
3
votes
0answers
306 views

Recognizing pure tensors in tensor product of vector spaces

Let $V$ be a vector space and let $\{e_i\}$ be a basis for it. Then $\{e_I\equiv e_{i_1}\otimes...\otimes e_{i_r}\}$ is a basis for $V\otimes ... \otimes V$. Suppose I am given an element $w=\sum a_I ...
0
votes
1answer
31 views

Is this proof that $\mathcal{L}(V_1,\dots,V_k;W)\simeq \mathcal{L}(V_1;\mathcal{L}(V_2,\dots,V_k;W))$ correct?

I've been trying to show that if $V_1,\dots,V_k,W$ are vector spaces over $K$, then $$\mathcal{L}(V_1,\dots,V_k;W)\simeq \mathcal{L}(V_1;\mathcal{L}(V_2,\dots,V_k;W)),$$ I think I've got the idea, ...
1
vote
1answer
283 views

Direct sum and tensor product of two representations of a group

Our lecturer gave us a hard exercice to go further in group theory (we stopped at group actions) : Let G be a group, V and W complex vector spaces and $\rho_1 : G \mapsto GL(V) $ be a group ...
3
votes
1answer
102 views

Existence of isomorphism between tensor products.

In multilinear algebra many maps are usually proven to exist rather than simply defined. For example, commutativity is one such example. In the book I'm studying the author says: let $V_1,\dots,V_k$ ...
8
votes
3answers
668 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
5
votes
2answers
277 views

Tensor Multiplication - Why should we use permutations?

I'm reading a book on multilinear algebra, and the author first establishes this easy isomorphism: if $V_1,\dots,V_k$ are vector spaces over the field $K$ and if $\sigma\in S_k$, then there is an ...
2
votes
3answers
173 views

How to really understand the tensor algebra?

If $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be $$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$ together with the ...
2
votes
1answer
65 views

Why it makes sense to think of multivectors as “paralelograms”?

Let $V$ be a vector space over the field $\mathbb{K}$ and let $T(V)$ be it's tensor algebra. We usually define the exterior algebra $\Lambda (V)$ by the following process: we consider the bilateral ...
2
votes
2answers
174 views

Why in differential geometry tensors are usually defined as multilinear maps?

In multilinear algebra books tensors are usually defined through the universal property. Given a family of $k$ vector spaces $V_1,\dots,V_k$ over the same field $F$ we want to construct a space $S$ ...
3
votes
1answer
179 views

Poincare duality and Hodge duality

The Poicare duality is defined in Greub's Multilinear algebra (1967) in Chapter 6, §2 as a isomorphism between $\bigwedge V$ and $\bigwedge V^*$, where $V$ is a finite-dimensional vector space, $V^*$ ...
1
vote
1answer
391 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
3
votes
4answers
386 views

What's a good reference to study multilinear algebra?

This semester I'm taking a course in linear algebra and now at the end of the course we came to study the tensor product and multilinear algebra in general. I've already studied this theme in the past ...
2
votes
2answers
206 views

p-forms as multilinear maps

I'm studying differential geometry and am learning about differential forms. We have a very intuitive and simple way to understand 1-forms as linear maps on from the tangent space to the base field, ...
3
votes
2answers
275 views

Eisenbud's proof of right-exactness of the exterior algebra

I'm trying to understand the proof in Eisenbud's Commutative Algebra that, given a right exact sequence $$K \to N \to M \to 0$$ of $R$-modules, we have an exact sequence $$K \otimes \wedge N \to ...
2
votes
0answers
302 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha ...
9
votes
2answers
842 views

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product ...
8
votes
0answers
381 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
0
votes
2answers
21 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
0
votes
1answer
182 views

Pairing of vector spaces

Suppose we have a pairing i.e. a bilinear map $\phi : V \times V \rightarrow \mathbb{R} $. There are many ways of getting a map $ \psi: V \rightarrow V^* $ two of which are $v\rightarrow \langle ...
1
vote
1answer
151 views

Lipschitz condition in infinite dimensional vector spaces

If we have that $T:V \times W \rightarrow Y$ multilinear and $V,W$ are infinite-dimensional normed vector spaces.(the finite-dimensional proof is easy, since you can use compactness of the boundary ...
1
vote
0answers
69 views

Describing multilinear maps as linear operators

Let a finite dimensional complex vector space $V$ be given. Let $T^k(V)$ denote the vector space of multilinear maps $V^k\to\mathbb C$. My original question was going to be as follows: Does there ...
3
votes
1answer
85 views

Finding all alternating bilinear $T$ that preserve a certain group of isometries of $\mathbb{R}^{n+1}$

Let $$G=\left\{\begin{pmatrix} H & 0 \\ 0 & 1\end{pmatrix} \ | \ H\in O(n), HJ=JH \right\}\subset \mathrm{Lin}(\mathbb{R}^{n+1},\mathbb{R}^{n+1}) $$ where: $n=2m$, $J$ is the standard complex ...
3
votes
2answers
64 views

A clarification regarding partial derivatives

Let us suppose the $i^{th}$ partial derivative of $f:\Bbb{R}^n\to \Bbb{R}$ exists at $P$; i.e. if $P=(x_1,x_2,\dots,x^n)$, $$\frac{f(x_1,x_2,\dots,x_n+\Delta x_n)-f(x_1,x_2,\dots,x_n)}{\Delta ...
1
vote
1answer
44 views

Terminology for a generalization of multilinearity to any algebraic structure.

Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto ...
1
vote
0answers
97 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
1
vote
1answer
74 views

Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many ...
9
votes
3answers
309 views

Polarization formula.

Let $V$ be a $\mathbb{R}$-vector space. Let $\Phi:V^n\to\mathbb{R}$ a multilinear symmetric operator. Is it true and how do we show that for any $v_1,\ldots,v_n\in V$, we have: ...
5
votes
1answer
184 views

Symmetric multilinear form from an homogenous form.

Let $V$ be a $n$-dimensional $\mathbb{R}$-vector space. Let $\phi:V\to\mathbb{R}$ a homogeneous form of degree $n$, i.e. $\phi(\lambda v)=\lambda^n \phi(v)$. If we define the symmetric multinear ...
4
votes
0answers
240 views

Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
0
votes
1answer
53 views

Determinant of symmetric matrix of the form $v\otimes v$

Note that for $V=\mathbf{R}^n$, $$S^2V = \{ v\otimes w \mid v, w\in V\text{ and }v\otimes w=w\otimes v \} =\{ A\in \mathrm{M}_2(\mathbf{R}) \mid A=A^T \}.$$ Clearly, $S^2V $ contains $O=\{ v\otimes v ...
2
votes
2answers
110 views

Universal property definition from Greub's Multilinear Algebra

Hi I started studying Greub's multilinear algebra book and I found something very strange when he defines the tensor product of two vector spaces: He defines: [...] Let $E$ and $F$ be vector spaces ...
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vote
0answers
31 views

Terminology for multilinear functionals the sum of whose cyclic shifts is zero?

Let $\varphi$ be an $n$-linear functional on a vector space $X$. Suppose that $\varphi$ has the property that, for all $(x_1,\ldots,x_n) \in X^n$, we have $$ \varphi(x_1,\ldots,x_n) + ...
5
votes
1answer
412 views

Trace of the $n$-th symmetric power of a linear map

Suppose $V$ is a vector space over $k$ and $\dim(V) = N$. Let $A \in\operatorname{End}(V)$. Let $\wedge^n A \in \operatorname{End}(\wedge^n V)$ where $\wedge^n$ is the $n$-th exterior power. I am ...
1
vote
1answer
36 views

Help with substituting definitions into tensor

I have 4 definitions for the following (Einstein summation) tensor $A^{ijk}A^{*}_{ijk}=A^{111}A^{*}_{111}+3(A^{112}A^{*}_{112})+3(A^{122}A^{*}_{122})+A^{222}A^{*}_{222}$ If I have these 4 ...
0
votes
1answer
122 views

Bilinear Form - Proof [duplicate]

I have to prove that the mapping $f(x,y) = {\displaystyle \sum_{i=1} ^ {n} }{ \displaystyle \sum_{j=1}^{n} }x_iy_j{f}(e_i,e_j)$ is a bilinear form, that is, inter alia, the condition: ...
1
vote
2answers
87 views

bilinear form - proof

I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$ is a bilinear form, that is, inter alia, the condition: ...
1
vote
1answer
87 views

Facts about the tensor product

I'm proving some statements about density operators, and would like to use two things. The problem is, that I'm not entirely sure they hold. Let $V,\; W$ be finite dimensional, complex inner product ...
2
votes
1answer
97 views

Linear map induced by bilinear maps

Suppose $f:X\times Y\rightarrow Z$ and $g:X\times Y\rightarrow W$ are bilinear maps in the category of vector spaces (say, real). Define the null space $N_1(f) :=\{x \in X: f(x,y)=0 \ \forall\, y\in ...
9
votes
2answers
704 views

Quick question: tensor product and dual of vector space

Recall that for a finite dimensional vector space $V$ we have the natural isomorphism $\phi :V^{*} \otimes V \rightarrow Hom(V,V)$ given by $\alpha \otimes v \mapsto (x \mapsto \alpha (x)v)$. Is ...