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

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Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
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40 views

Why are vectors considered to be rank (0,1) tensors and dual vectors considered to be rank (1,0) tensors?

Sean Carrol in his book of general relativity, he defines a tensor to be a multilinear map from a collection of dual vectors and vectors to $\mathbb{R}$: $T:T^*_p \times...\times T^*_p \times T_p ...
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1answer
32 views

Multilinear Mappings

Let $E$, $F$ complex Banach spaces and $p,q\in \mathbb{N}$ with $p+q\geq 1$. I will denote by $\mathcal{L}_a(^{p,q}E;F)$ the subspace of all $(p+q)$-linear mappings $A\in ...
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1answer
63 views

Trace of the $k$-th Exterior Power of a Linear Operator

Let $V$ be an $n$ dimensional vector space over a field $F$ and $T$ be a linear operator over $V$. Assume that the characteristic of $F$ is not $2$. Definition. Consider the map $f_1:V^n\to ...
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Cannonical forms for symmetric tensors of type (2,1)

Let $$T:V\times V \rightarrow V$$ be a symmetric (2,1) tensor, that is $$ T(X,Y)=T(Y,X) \,\ \forall X,Y \in V.$$ In the simple case when $\dim V=2,$ there is always a frame $F$ of $V$ , $F=(e_1,e_2)$ ...
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27 views

Universal Linearizer of Alternating Multi-$F[x]$-Linear Maps is Same as that of Multi-$F$-Linear Maps.

Let $V$ be a an $n$-dimensional vector space over a field $F$. Let $M=F[x]\otimes_F V$. We can consider $M$ as an $F[x]$-module by extending scalars using the inclusion $F\to F[x]$. Fact 1. There is ...
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A question about functions $f : \mathbb{R} \leftarrow \mathbb{R}$ such that $\mathop{\lambda}_{x,y:\mathbb{R}}f(x+y)-f(x)-f(y)$ is multilinear.

Suppose $f : \mathbb{R} \leftarrow \mathbb{R}$ is one of the following two functions: $$f(x) = x^2, \qquad f(x) = x$$ Then: $f$ is a monoid homomorphism of the multiplicative structure of ...
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1answer
50 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
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76 views

Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
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21 views

Equality of two multilinear forms

Take two multilinear forms $f,g$ defined on the same set $E$ such that $\forall x\in E,f(x,x,\dots,x)=g(x,x,\dots,x)$. Does that imply that the two functions are necessarily equal ? I can't seem to ...
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1answer
70 views

The set of all bounded multilinear maps

I'm not sure how to show that the set of all bounded multilinear maps is a vector space. Could someone help me?
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1answer
31 views

Canonical forms for tensors of type (2,1)

Are there any canonical forms for tensors of type (2,1)? Such a tensor can be defined as a bi-linear map $$ T:V \times V \rightarrow V,$$ for $V$ a finitely dimensional real vector space.
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1answer
27 views

Is there an invariant similar to the characteristic polynomial for (0,2) and (2,0) tensors?

The characteristic polynomial of a matrix - a (1,1) tensor - is its invariant (independent on basis transformation). Is there a similar invariant for (0,2) and (2,0) tensors? The characteristic ...
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1answer
32 views

Tensor Contraction Invariance

On page 86 of Bishop and Goldberg's Tensor Analysis on Manifolds you are asked to show that contractions are invariants. Rather than doing it awkwardly for an (r,s)-tensor (r is the contravariant and ...
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2answers
58 views

Multilinear and alternating property of $\det(f)$ where $f$ is an endomorphism

Everybody knows the determinant of a matrix $A\in k^{n\times n}$ ($k$ a commutative ring) and everybody knows that the determinant of $A$ is an alternating multilinear map in the columns aswell as in ...
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18 views

#skew symmetric, symmetric and alternating multilinear map in a vector space over field of characteristic 2

can every skew symmetric multilinear map written as sum of symmetric and alternating multilinear map.(specially I mean in a field of characteristic 2)
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45 views

Decomposition into simple bivectors

According to Wikipedia, any element of $\wedge^2\Bbb R^n$ should be decomposable into $n/2$ simple bivectors for $n$ even or $(n-1)/2$ for $n$ odd. How do I count that? How do I check that ...
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29 views

2 forms and Base

Let$\: V \;$ be a n-dimensional vector space and $\:w\;$ a two form. Proof that there exists a base $\alpha_1,\alpha_2,..\alpha_n, \in V^* \;$ so that $\; \omega =\alpha_1 \wedge \alpha_2 + \alpha_2 ...
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1answer
56 views

Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity. I am ...
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1answer
38 views

Confusion regarding notation of a dual transformation

I'm reading Spivak's Calculus on Manifolds and in Chapter 4 he defines the dual transformation (although he doesn't call it that) as follows: If $f:V \rightarrow W$ is a linear transformation, a ...
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1answer
47 views

Generalization of scalar product for vectors in n-dimensional space

Let $x$ and $y$ be two vectors and $A$ the angle between them. Then we have the scalar product $$x\cdot y = \|x\|\|y\| \cos A$$ Let $x$, $y$ and $z$ be three vectors; $A$ angle between $x$ and $y$; ...
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32 views

General definition of free module over arbitrary rings and space of multi-linear functions

Consider definition of free module as given in this question: Free modules over commutative ring (possibly without unity) where free means having a LI spanning set I have proved if Can we say if V ...
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2answers
37 views

Dual space of exterior power and exterior power of dual space

Let $V$ be a finite-dimensional vector space. Is there an isomorphism between $\Lambda^k(V^\ast)$ and $\left(\Lambda^k(V)\right)^\ast$? I was able to prove this with the additional requirement of ...
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1answer
55 views

Map to exterior power gives rise to smooth embedding of Grassmannian in projective space?

How do I see that the map$$(x_1, \dots, x_n) \mapsto x_1 \wedge \dots \wedge x_n$$from $V_n(\mathbb{R}^m)$ to the exterior power $\wedge^n(\mathbb{R}^m)$ gives rise to a smooth embedding of ...
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28 views

Prove that every polynomial from a finite Banach space to another Banach space has an unique representation in terms of coordinate functionals

I'm trying to solve Problem 1.2.J in Mujica's "Complex Analysis in Banach Spaces". The problem states as follows: Let $E$ and $F$ be Banach spaces over $\mathbb{K}$, with $E$ finite dimensional. ...
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15 views

Block-matrix constructions for outer product of sets of vectors and higher order tensors.

So I'm in the process of doing a large sequence of vector-vector outer-products. On my software it would be convenient to do this in a systematic vectorizable way. Say I have my $n$-dimensional column ...
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47 views

Meaning of / intuition for contraction of tensors (in the Riemannian setting)

I'm currently taking a course in differential geometry, and we are, I'm guessing, finally going to start working with the Riemannian curvature tensor after having covered a lot of smooth manifold ...
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1answer
87 views

A coordinate free book on linear and multilinear algebra defining determinants using exterior algebra

I would like to find an advanced introduction to linear and multilinear algebra that is 1)Coordinate free 2)Use tensor products and exterior algebras to define determinants 3)DOES NOT assume a ...
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1answer
61 views

On volume forms and norms on exterior powers

Let $V$ be a $2$-dimensional vector space. Given an inner product on $V$ one may define an inner product on the simple $k$-vectors of $\Lambda^k(V)$ by $$\langle v_1 \wedge \cdots \wedge v_k, w_1 ...
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1answer
43 views

How can I prove that the norm of the following operator is $\frac{1}{m!}$?

I'm trying to prove that the norm of the multilinear symmetric operator $A$ is $\frac{1}{m!}$ where $A$ is defined as: $$ A(x_1,\dots, x_m) = \frac{1}{m!} \sum_{\sigma \in S_m} \xi_1(x_{\sigma(1)} ) ...
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27 views

Erdos-Ko-Rado theorem when k divides n

So i'm trying to prove Erdos-Ko-Rado for a Kneser graph $K(k,n)$ when I know that $k$ divides $n$. If $C$ is some coclique of $K(k,n)$ I've argued that for any ordered partition $(A_1,..,A_{n/k})$ of ...
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1answer
47 views

Integral of exponential with linear term

$$x \in \mathbb{R^n},$$ M is a positive symmetrical nonsingular nxn Matrix and j is an arbitrary vector in $$\mathbb{R}^n.$$ The following has to be calculated: $$Z(j) = \int_\mathbb{R^n} ...
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1answer
29 views

cardinality of orbits

Let $G=\mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )\times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p ) \times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )$, $p$ prime, act on $(\mathbb{Z}/p\times ...
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Concrete description of altenating or symmetric power functor (analogous to Kronecker product)

Suppose that an $A$-module homomorphism is given between two free and finite dimensional $A$ modules $M$ and $N$, with basis $e_1 \ldots e_n $ and $f_1 \ldots f_m$ respectively, by a matrix $T \in ...
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1answer
34 views

Understanding a step in a computation involving dual basis and permutations

Since $\varphi_{i_j} \in \mathcal{T}(\mathbb{R}^n)$, for every $j = 1, \dots, k$, we have \begin{align*} \varphi_{i_1}\wedge\dots\wedge\varphi_{i_k}(e_{i_1}, \dots, e_{i_k}) &= ...
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1answer
169 views

Proving that $\phi$ is an alternating multilinear map

Let $V$ be a finite-dimensional vector space over the field $F$ with basis $\mathcal B = \{v_1,\dots,v_n\}$. Let $1\leq k\leq n$ and pick some $1\leq i_1 < i_2 < \dots < i_k \leq n$. I am ...
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1answer
25 views

Manipulation of skew-symmetric linear map

Let $\Delta$ be a skew-symmetric $n$-linear map. I have the following in my notes and I am having trouble seeing how it follows: $$ \Delta\left(\sum_{i=1}^n{e_i}, \sum_{i=1}^n{(e_i)} -e_2, ...
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1answer
40 views

Differential and Rank of $XAX^{-1}-A$

I have a map: $F_{A} (X) :GL\left(2n,\mathbb{R}\right) \longrightarrow\mathbb{\mathfrak{M}_{\mathit{2n\times2n}}\left(\mathbb{R}\right)}$ such as \begin{eqnarray} & F_{A}(X) & =XAX^{-1}-A ...
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18 views

Trace form seen by tensorial product.

I'm studying tensor product by the book "Multilinear Algebra - W. H. Greub". On page $38$ I couldn't understand how he got into the equation $(1.41)$. Could anyone help me?
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1answer
60 views

Generalized “scalar product” based on multilinear form?

In an $\mathbb{R}$-vector space $V$, the scalar product is a paradigmatic example of a non-degenerate, symmetric, positive-definite bilinear form $\beta : V \times V \to \mathbb{R}$. I wonder if the ...
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37 views

Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
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1answer
51 views

$M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?

Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if ...
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1answer
54 views

Signature of the inner product.

I need some help with this problem. Consider in $\mathbb{R}^{4}$ the lorentz's metric $h$ which has a signature $(3,1)$, this mean that diagonal matrix $M(h)$ is the form $$M(h)=\begin{pmatrix} 1 ...
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1answer
23 views

derivative of kronecker product

Given $x \in \mathbb{R}^N$ and a function $$H = \sum_{i,j,k=1}^n\ J_{i,j,k}\ x_i x_j x_k$$ for a fixed $J \in \mathbb{R}^{n \times n \times n}$, I am trying to calculate the derivative $\frac{d H}{d ...
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Grammatically confused: $\omega=4dV$ for 3-form $\omega$ and volume in $\Bbb R^4$?

Background: Against the advice I should have been given but wasn't, I'm taking a Lie theory course with no background in differential geometry. We finally made it into the part of the course where we ...
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26 views

Composition of linear maps as a tensor product

$\mathbf{Question:}$ Let $V$ be a finite-dimensional vector space over an algebraically closed field $F$. Fix $A,B \in \mathscr{L}(V)$. Consider the linear operator $T_{A,B} \in ...
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1answer
47 views

How to understand acting one tensor on another tensor to obtain a third tensor?

I've already known the definition of the tensor that a tensor T of type $(k,l)$ is a multilinear map from a collection of dual vectors and vectors to $\mathbf{R}$: $$ T: ...
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1answer
30 views

Dimension of the basis $\{x \otimes y + y \otimes x\}$

I'm trying to prove that the annihilator of $I = \left<x \otimes y - y \otimes x \right>$ is $\left<x \otimes y + y \otimes x \right>$. To do this I am trying to compare dimensions. So if ...
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29 views

Problem with triple cross product proof

When trying to prove bac-cab rule, I get to point, where I don´t know, what is true. I have $$ (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_jb_lc_m=a_jb_ic_j-a_lb_lc_i $$ but when $$ ...
3
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1answer
49 views

raising/ lowering indices

Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ...