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

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21 views

Intrinsic Definition of $^\sigma\alpha$, where $\alpha$ a covariant $k$-tensor and $\sigma\in S_k$

Let $V$ be a finite dimensional vector space and $\mathcal T^k(V^*)$ be the set of all the covariant $k$-tensors on $V$. The symmetric group $S_k$ acts on $\mathcal T^k(V^*)$ as follows: Given ...
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0answers
21 views

find value of y, which depends linearly on variables, x and z. [on hold]

I have a doubt, something very new to me and dont know whom to ask, so shouting out over here. My problem is , there is a function f(y) which is linear to variable x, eg. f(y) = ax + b; And also f(y) ...
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1answer
32 views

What is the prerequisiste to study Tensors for application in signal processing?

I want to study Blind Source separation in signal processing for this I need to study Tensors and have a basic idea about rank, border rank and other concepts. Right now I am studying from ...
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1answer
27 views

Some detail needed in Positive Definite Matrix

First all of all, I am sorry I have put a page of my text book. Somehow I need some help to understand some paragraphs in the page. If you can explain, please let me have some explanations to those in ...
2
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3answers
38 views

Linear Algebra Question for Repeated Eigenvalues

Can somebody explain what the sentence in the red circle means?
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0answers
11 views

use universal properties to prove the existence of isomorphism

Use universal properties to prove that for a finite dimesional vector space $V$ and $W$ there is a canonical isomorphism: $$\bigwedge^2(V\oplus W)\to \bigwedge^2V\oplus(V\otimes W)\oplus\bigwedge^2W$$ ...
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1answer
49 views

How to prove that below quantity is a Third Rank Tensor

$F^{ik}$ is an antisymmetric tensor. I want to prove that below quantity is a Third Rank Tensor. $$\dfrac{\partial F_{ik}}{\partial x^{l}} + \dfrac{\partial F_{kl}}{\partial x^{i}} + \dfrac{\partial ...
2
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0answers
33 views

$L\cdot M:=\pi_{r+s}(L\otimes M)$, then $((f_i\cdot f_j)\cdot f_k)\cdot f_l=?$

I'm reading Kenneth Hoffman's "Linear Algebra", Ed 2. In $\S5.7$ "the Grassman Ring" it tries to explain the way lead to the definition of exterior product (a.k.a. wedge product). However I got some ...
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0answers
17 views

on “strong positive semi-definiteness”

Suppose $M_{ij}^{kl}$ is a real symmetric $nm \times nm$ matrix ($0 \leq i,j \leq n$ and $0 \leq k,l \leq m$), i.e. $\forall i,j,k,l$ \begin{equation} M_{ij}^{kl} = M_{ji}^{lk}. \end{equation} Can we ...
2
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1answer
26 views

From linear transformation to alternating linear transformation

I'm reading Kenneth Hoffman's Linear Algebra, Ed2. In $\S5.6$ "Multilinear Functions" it talks about generating an alternating linear transformation from a linear transformation. The collection ...
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0answers
34 views

Why are free objects related to tensors important?

Let $R$ be a commutative ring and $M$ be an $R$-module. Then, we can construct the tensor algebra $T(M):=\oplus_{n\in \mathbb{N}} M^{\otimes n}$ which is likely a free object. That is, for any ...
3
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1answer
30 views

Is the symmetric algebra direct sum of $k$-th symmetric powers?

Let $R$ be a commutative ring and $M$ be an $R$-module. Let $S(M)$ be the symmetric algebra of $M$ and $S^n(M)$ be $n$-th symmetric power of $M$. Then is $S(M)$ the internal direct sum of $S^n(M)$? ...
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0answers
31 views

What do we call $R$-bilinear maps $V,W \rightarrow R$?

Let $R$ denote a fixed commutative ring. Then given an $m \times n$ matrix $A$ with entries in $R$, we get an $R$-linear transform $R^n \rightarrow R^m$ in the usual way. We also get an $R$-bilinear ...
3
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1answer
48 views

Plücker Relation: misunderstanding?

I'm trying to understand exterior algebra better by gaining some "bare hands" understanding of the exterior powers $\Lambda^k(X)$ in more detail when $\dim(X)$ is small. I think so far I understand ...
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0answers
32 views

Tensor Invariants

I was learning about tensors and vectors and it was mentioned in class that they were "invariant" objects that take on specific representations in different bases. I get this for vectors -- I can ...
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3answers
98 views

Is it true that every element of $V \otimes W$ is a simple tensor $v \otimes w$?

I know that every vector in a tensor product $V \otimes W$ is a sum of simple tensors $v \otimes w$ with $v \in V$ and $w \in W$. In other words, any $u \in V \otimes W$ can be expressed in the ...
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1answer
35 views

Exterior power of multilinear functions applied to linearly dependent vectors is zero

I'm working on a homework problem, and we are to show that if $T \in \wedge^p V^*$, and $v_1,\ldots,v_p$ are linearly dependent, then $T(v_1,\ldots,v_p) = 0$. What I've got so far: I understand that ...
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0answers
10 views

Find normal random variables from independent standard normal variables with correlation matrix

I'm trying to find three independent standard normal variables from three normal random variables using a correlation matrix. So far, I have decomposed the problem using the matrix's cholesky ...
17
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4answers
411 views

Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
4
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2answers
146 views

Basis for Tensor Product of Infinite Dimensional Vector Spaces

If V and W are vector spaces over a common field with bases $V_B = ${$v_i : i \in I$} and $W_B = ${$w_j : j \in J$}, then is {$v_i \otimes w_j: i \in I, j \in J$} a basis for $V \otimes W$ ? I have ...
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1answer
28 views

What is this map between tensor spaces called? (Change of coordinates).

Let $V$ and $W$ be finite dimensional vector spaces. Let $A: V \to W$ be linear. Define the map $A^* : (W^*)^{\otimes r} \to (V^*)^{\otimes r}$ by $(A^*\alpha)(w_1, w_2, \ldots, w_r) = \alpha(Aw_1, ...
0
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1answer
25 views

Define $f: V \times V \rightarrow \mathbb{K}$ ($\mathbb{K}$ field), by $f(u, v) = \langle u, Av\rangle$

Define $f: V \times V \rightarrow \mathbb{K}$ ($\mathbb{K}$ field), by $f(u, v) = \langle u, Av\rangle$, any $u$, $v \in V$ and $A \in M(n, \mathbb{R})$ with $A^T = -A$, i.e., $A$ is skew-symmetric. ...
2
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0answers
39 views

Natural Isomorphism between $V^*\otimes W^*$ and $\mathcal L^2(V,W; F)$.

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and ...
4
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1answer
61 views

Invariants of $V^{\otimes N}$. [closed]

Let $V$ be a finite dimensional complex vector space, and $G = SL(V)$ be the group of linear transformations of $V$ with determinant $1$. (a) Show that $V^{\otimes N}$ contains a nonzero ...
2
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1answer
49 views

why integrating only alternating forms?

Hello I was reviewing some concepts of differential forms. I cannot recall why only multilinear alternating forms can be integrated on manyfolds and not general multilinear forms... Why is the ...
1
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3answers
72 views

Why is quadratic form defined via a symmetric bilinear form?

A typical definition of quadratic form goes like this: Let $B:V\times V \to F$ be a symmetric bilinear form. A function $Q : V → F$ defined by $Q(v) = B(v, v)$ is called a quadratic form. Why ...
1
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1answer
25 views

Isomorphism between different definitions of symmetric tensors

Hi I was working through symmetric tensors (technically in a differential manifold, but we defined it abstractly). We defined them first in the usual way: Consider $A$ an integral domain and ...
3
votes
2answers
49 views

To show $L\otimes_K \text {End}_K(V)\cong \text{End}_L(L\otimes_K V)$

Let $L/K$ be field extension and let $V$ be a $K$-vector space. Then do we have an isomorphism $$L\otimes_K \text {End}_K(V)\cong \text{End}_L(L\otimes_K V)$$ as $L$-algebras? My attempt: for ...
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0answers
46 views

Introduction to tensor (for graph analysis)

I am starting a PhD program in social network analysis and I would like to have some suggestions about introductory books and online material to Tensor calculus. I am a complete newbie in algebra. ...
0
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0answers
22 views

Help in the proof of Poincaré Theorem to differential forms

I'm revising the proof of Poincaré Theorem, but I don't understand a pass of proof. Let be $E$ and $F$, Banach spaces and $U\subset E$ open set. Consider $\omega\in\Omega_p^n(U;F)$ a p-differential ...
0
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1answer
34 views

How to prove some formulations aboult Kronecker product?

The Kronecker product has some properties as the wikipedia http://en.wikipedia.org/wiki/Kronecker_product. For the sake of simplicity, we denote $\mathbf{U}_{M}^T\otimes ...
2
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0answers
52 views

Eigenvalues of a Kronecker Product type matrix

We have matrix $C$ of the form: $C =\begin{bmatrix} B_{1,1} A_{1,1} & B_{1,2} A_{1,2} & \dots & B_{1,K} A_{1,K} \\ B_{2,1} A_{2,1} & B_{2,2} A_{2,2} & \dots & B_{2,K} ...
2
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0answers
44 views

How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?

The mode-$n$ product of a tensor $\mathcal{X}=[x_{i_1,\ldots,i_M}]\in \mathbb{R}^{I_1\times \cdots \times I_M}$ and a matrix $\mathbf{U}=[u_{i_m,j}]\in \mathbb{R}^{I_m\times J}$ is denoted by ...
0
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0answers
36 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...
1
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1answer
35 views

Show that the trace of the operator $S \wedge T$ is zero

I have some difficulties with the following problem: Let $V$ be a finite dimensional vector space over $\mathbb{K}$. Let $S,T \in L(V,V)$. Show that the trace of the operator $S \wedge ...
1
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2answers
62 views

Question about tensor products, decomposable tensors, …

I need some help with the following problem: Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$. Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$. ...
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1answer
28 views

Relationship with trace and asymptotic stability in control theory [closed]

What is the relationship between $\mathrm{tr}(\exp(tA) \exp(tA^\ast))$ and asymptotic stability in control theory ?
6
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1answer
79 views

Differentiation on Manifolds Basics

I'm having some real trouble comprehending integral curves and Lie derivatives on a Manifold. I will write out my understanding and ask the questions below. For a vector field $X$ on smooth manifold ...
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1answer
43 views

$(2n-1)$-form is closed [closed]

Consider in $\mathbb{R}^{2n}$ differential forms$$\omega = dx_1 \wedge dx_2 \wedge \dots \wedge dx_n\text{ and }\theta = \sum_{n+1}^{2n} (-1)^{j-1}x_jdx_{n+1} \wedge \dots \wedge dx_{j-1} \wedge ...
1
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1answer
55 views

Tensor product definition?

I am getting a bit confused on the notation used for tensor products, is we have the tensor product space $V\otimes V^*$ if $v\in V$ and $a \in V^*$ then is the following correct? $$v \otimes ...
7
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1answer
108 views

Tensors as Multilinear maps?

Today I learned about Tensors as multilinear maps. I usually think of tensors as a multidimensional array of numbers with fixed transformation laws, and I am having trouble understanding how tensors ...
7
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1answer
188 views

Eigenvalues of Kronecker Product

Maybe it's simple but I can't see the solution of this problem (Russell Merris, Multilinear Algebra, CRC Press, 1997, chapter 6, p.202, exercise 4): Let $\lambda_1,\ldots,\lambda_p$ be the ...
1
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1answer
36 views

Defining addition of vectors of different dimensions

While doing real data analysis I came up with a problem. I have given lots of efforts to solve it and could not succeed. Here is the problem: Suppose, we have a set of vectors ...
2
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0answers
49 views

Finding a maximal isotropic subspace

I have the following question: Let $V$ be a finite dimensional complex vector space. For a given bilinear form $(,): V \times V \rightarrow \mathbb{C}$, a subspace $W$ of $V$ is called isotropic with ...
2
votes
1answer
52 views

Inverse of covariant tensor of rank two is contravariant.

I'm studying tensors on my own, using "Tensor Calculus" from David C. Kay, and there is this theorem in page $29$: Suppose that $(T_{ij})$ is a covariant tensor of order two. If the matrix ...
1
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1answer
27 views

Multilinear maps: is $\phi(av_1,v_2)$ always equal to $\phi(v_1,av_2)$?

I am learning about multilinear maps by myself and the book I'm following gives a definition which is somewhat vague. That's the definition: Given vector spaces $V_1,V_2,\dots,V_p,W$. A mapping ...
2
votes
1answer
58 views

Multilinear algebra some basics.

The wedge product of $p$ vectors in vector space $V$ is called a $p$-vector and the vector space generated by all $p$-vectors is denoted $\bigwedge^p V$ with the basis $e_I:=e_{i1}\wedge\dots\wedge ...
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1answer
160 views

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions?

How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? Needed for the irrep decompositon of $3\otimes 3\otimes 3$ in here. No idea where to start to ...
1
vote
1answer
62 views

acrobatics with $2$-form in $\mathbb{R}^{2n}$ [closed]

In the space $V = \mathbb{R}^{2n}$ with coordinates $(x_1, \dots, x_n, y_1, \dots, y_n)$ consider the $2$-form $\omega = \sum_{i=1}^n x_i \wedge y_i$. Let $A$ be a $n \times n$ matrix. Consider a ...
7
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6answers
144 views

coordinate free proof that $\text{div}(\nabla f \times \nabla g) = 0$

Let $V$ be a Euclidean $3$-dimensional space. Does there exist a coordinate-free proof that for any two $C^1$-functions $f, g: \mathbb{R}^3 \to \mathbb{R}$ we have $$\text{div}(\nabla f \times \nabla ...