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

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1answer
25 views

If a Bilinear Form is Non-Degenerate on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
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1answer
27 views

Representing a series of Matrix inner product with a single matrix product.

I have a set of constraints in my optimization problem, constraints in the form , $\langle A, e_i e_j^T \rangle = r_{ij} ,\forall i,j \epsilon S$, where $A$ is an $n*n$ semidefinite and symmetric ...
4
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1answer
70 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
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0answers
15 views

The space $V^{0}_{p}$ of p times covariant tensors and canonical isomorphisms

I have been studying tensor calculus by myself, but I have found the following claim in my book: The space $V^{0}_{p}=V^{*} \otimes \cdots \otimes V^{*}$ of $p$ times covariant tensors is ...
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1answer
28 views

Have a question about Linear Transformations

Explain why there cannot be a linear transformation T: $R^2$ --> $R^2$ for which T(1,1)=(2,3) and T(3,3)=(1,4). I have no clue how to start this problem. Wouldn't ...
4
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0answers
49 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
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1answer
23 views

What is the best fit (in the sense of least-squares) to the data?

A) Find the best fit (in the sense of least-squares) to the data $x_1$ $(1,-1,-1,1)$ $x_2$ $(1,1,-1,-1)$ $y$ $(5,1,1,1)$ by a linear function of the form $y$=$a$+$bx_1$+$cx_2$ B) Find ...
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0answers
18 views

Solving a linear equation with a modulus

I'm working through http://www.reteam.org/papers/e59.pdf I'm really having difficulty with this section. Since we now have m, we can easily set up a linear system of equations which solves ...
8
votes
2answers
35 views

$\text{Alt}\,(\phi_1 \otimes \phi_2 \otimes \phi_3)$

How do I write out $\text{Alt}(\phi_1 \otimes \phi_2 \otimes \phi_3)$ for $\phi_1, \phi_2, \phi_3 \in V^*$?
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2answers
33 views

Contraction as Adjoint of Wedging

Let $V$ be an $n$-dimensional vector space. Given $\phi^1\wedge \cdots \wedge \phi^k\in \bigwedge ^k(V^*)$ and $v_1\wedge\cdots\wedge v_k\in \bigwedge^k(V)$, we write $$ \langle \phi^1\wedge ...
7
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1answer
208 views

Symmetric kernel of tensor product

Let $V,W$ be two real vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with distinct kernels $K_i$ of dimension $1$. Consider the tensor product of these maps ...
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0answers
17 views

if |A|'s power m=1 then |A|=+1 or |A|=-1 ; for all m is +ve integer

I'm the student of software engineering and i wanted to get the mathematical proof of "if |A|'s power m=1 then |A|=+1 or |A|=-1 ; for all m is +ve integer" please response as soon as possible. ...
3
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1answer
47 views

Is there an “internal” definition of the tensor product?

We have the following "internal" definition of the direct sum: A vector space $V$ with subspaces $S,T$ is said to be the direct sum of $S$ and $T$ if $S + T = V$ and $S \cap T = \{0\}$. (Of course ...
0
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1answer
51 views

Complex functions and $\star_3$

I wanted to maybe extend Hodge star/ Technical question to a new question so others could benefit from the idea. So there we discussed that when the $\star$ is Hodge duality star then it is ...
0
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0answers
22 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 ...
0
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1answer
40 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 ...
1
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1answer
30 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
41 views

Linear Algebra Question for Repeated Eigenvalues

Can somebody explain what the sentence in the red circle means?
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0answers
16 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
54 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 ...
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0answers
34 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
18 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
27 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
35 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
38 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
35 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
54 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
34 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 ...
8
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3answers
106 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
39 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
11 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 ...
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4answers
428 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
167 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 ...
0
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1answer
30 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
27 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. ...
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0answers
42 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
63 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 ...
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1answer
51 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 ...
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3answers
83 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 ...
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1answer
26 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
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2answers
51 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
51 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
27 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
39 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 ...
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0answers
71 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} ...
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0answers
45 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 ...
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1answer
40 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
65 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 $Im(\varphi)=U$. ...
1
vote
1answer
30 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 ?