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

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10 views

common factors of multilinear polynomial

Say $F,G\in\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ are two multilinear polynomial. If $F$ and $G$ vanish at a common set of coordinantes $(a_{i1},a_{i2},\dots,a_{in-1},a_{in})\in\Bbb R^n$ for ...
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18 views

On Schur map and tableaux

My post refers to Jerzy Weyman's "Cohomology of vector bundles and syzygies" pag. 37. Let $R$ be a ring and $E$ a free $R$-module of rank $n$. Let ${e_1, \cdots, e_n}$ a basis of $E$. Let us consider ...
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1answer
56 views

Decompose $\omega:= e_0\wedge(e_1\wedge e_2 + e_3\wedge e_4)$

$(e_1\wedge e_2 + e_3\wedge e_4)$ is well-known to be of rank 2 (can't be decomposed). On the other hand, $\omega \wedge \omega = e_1\wedge e_1 \wedge ... = 0$. According to the wikipedia article ...
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1answer
22 views

Correct definition of bilinear(multilinear) maps over noncommutative rings

I wonder about the following: Let $R$ be a noncommutative ring with $\alpha, \beta \in R$ such that $\alpha \beta \neq \beta \alpha$. Let $M,N,P$ be left-$R$-modules. Why does it make problems to ...
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35 views

Do we need any special justification for concluding $f=0$?

Today, we were showing the following in class: Let $V$ be a finite-dimensional vector space over $F$. Prove that the bilinear map $$\beta:V \times V^* \to F, \ \ (v,f) \mapsto f(v)$$ is ...
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38 views

Showing $\det(I_E+f)=\sum_{p=0}^{n}\mathrm{tr}(f^p)$ where $f^p:Λ^{p}(E) \to Λ^{p}(E)$ and $f:E\to E$

$E$ is a vector space with dimension $n$ and $f:E\to E$ is a linear map and for every $p=1,2,3,...$ we have $f^p:Λ^{p}(E) \to Λ^{p}(E)$ which is defined as below ...
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45 views

$(n-1)$-alternative tensor on E are decomposable

$E$ is a real vector space with dimension $n$ and $E^*$ is dual space of $E$. Assume $\alpha \in Λ^{n-1}(E)$ Show that there exists $\alpha_1,\alpha_2,...,\alpha_{n-1} \in E^*$ such that ...
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1answer
61 views

Almost complex structure which fails to be compatible

Let $V$ be a real vector space equipped with a scalar product $\langle, \rangle$ (i.e. a positive definite symmetric bilinear form). We say that an endomorphism $J: V \to V$ is an almost complex ...
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1answer
70 views

How does the representation of co-vectors change if we change the basis of a vector space $V$?

I'm trying to understand how vectors, differential forms and multi-linear maps in general transform under change of coordinates. So I start with the simplest case of vectors. Here's my own attempt, ...
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19 views

symmetric bilinear form on $\mathbb{Z}_2$

Find all symmetric bilinear forms of a vector space $V$ of finite dimension on $\mathbb{Z}_2$. As every bilinear form is represented by a matrix then the idea is to find the set of matrices ...
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1answer
40 views

Is $\psi (x_1,…,x_n)=\det \begin {pmatrix} x^1_1 & \dots &x^1_n \\ \vdots & &\vdots \\ x^n_1 & \dots & x^n_n\end{pmatrix}$ multilinear?

Suppose $\forall x_1,\ldots,x_n,\in R^n$, denote $x_1=(x_1^1,x_1^2,\ldots,x_1^n),x_2=(x_2^1,\ldots,x^n_2)$. Define $\psi:V\times V\times \dots\times V\to R$ as follows: $$\psi (x_1,\ldots,x_n)=\det ...
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55 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
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48 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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1answer
74 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...
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51 views

Tensor analog of Matrix Product

Given two $n \times n$ matrices $A$ and $B$, we can form their matrix product in the usual way. Is there a similar product for tensors? E.g., if one is given two $n \times n \times n$ tensors ...
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1answer
26 views

Induced Action of Matrix on Tensor Product

I've been asked to write the induced action of a matrix in $M_4(\mathbb{R})$ on $\mathbb{R}^4 \otimes \mathbb R^4$, but this terminology is unfamiliar to me. What does it mean for a matrix to induce ...
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1answer
63 views

$G$-representations, $W \otimes V^* \to \text{Hom}(V,W)$

Let $V$ and $W$ be finite-dimensional vector spaces. I know how to construct an explicit isomorphism of vector spaces $W \otimes V^* \to \text{Hom}(V,W)$ and show that it's an isomorphism. But if I ...
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41 views

Do Tensors have a determinant property?

We know that only square $n \times n$ matrices have a determinant property! And it can be defined just like this: $$A=\begin{array} & & & \\ ...
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1answer
34 views

What is explicit isomorphic map between $\Bbb R$ and $\text{Alt}^n(\Bbb R^n)$

We denote $V \subset \mathbb{R}^n$ to be an open set. Definition 1: Let $\text{Alt}^p(\Bbb R^n)$ denote the set of all alternating multlinear functionals on $\Bbb R^n$. Definition 2: Let ...
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24 views

Multilinear Function proof in Spivak?

Note that $$\wedge^n (V)$$ denotes the set of all alternating multilinear functions and $\mathfrak{I}^n(R^n)$ denotes the set of all multilinear function. I don't know what the actual symbol ...
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33 views

A matrix equation with real coefficients

The problem is the following: Find $\lambda$ such that $ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $ where ...
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1answer
30 views

Multilinear algebra and matrices

Given $\wedge^k(V)$ an alternating multilinear space and $T : V \to W$ a linear map, then we have $$v_1 \wedge \dots \wedge v_k \in\wedge^k(V).$$ Define $$\wedge^k(T)(v_1\wedge\dots\wedge v_k) = ...
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83 views

How can I determine the number of wedge products of $1$-forms needed to express a $k$-form as a sum of such?

This question was motivated by this related one: How "far" a differential form is from an exterior product . Let $\mathbb{V}$ be a vector space of dimension $n$ with underlying field ...
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2answers
79 views

Help with a paper about tensors

I came across something in a paper I am not able to understand jet. Unfortunately the author is kind of short with explanations. Maybe someone here can help me to understand this. $M^d \in ...
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25 views

Indices question in this multilinear algebra question.

Suppose $V$ is finite dimensional with $\dim V = n$ and $f : V \to V$ be linear. Prove that there is a number $d(f)$ such that $$\Omega^n(f)(\omega) = d(f)\omega.$$ Here, $\Omega^n(V)$ denotes the ...
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2answers
72 views

Multilinear Algebra, finding $z \wedge z.$

Define $A^k(V)$ to be the set of all alternating multilinear functions from $V^k \to \mathbb{R}$. Consider the space $A^2(\mathbb{R}^4)$, does there exists $z\in A^2(\mathbb{R}^4)$ such that $z ...
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1answer
38 views

Polynomial and super-symmetric tensor

A quadratic function uniquely determines a symmetric matrix. Ok that’s easy. Now a homogeneous polynomial function $f(x)$ also uniquely determines a super-symmetric tensor. My question is how do I ...
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24 views

How to write an analogue to matrix-vector multiplation with an extra dimension in tensor notation

My background is severely lacking in tensor algebra, and after a few days of looking into tensors I am still not able to even formulate this question quite correctly; my apologies for that. I am aware ...
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1answer
22 views

Intersection of two sets

Let $E\subset{\mathbb R}^n$ be a set of the type $I_1\times \dots \times I_n$, where $I_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is ...
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123 views

Exploiting structure in multilinear equations

I'm wondering if there are any standard techniques for exploiting structure in multilinear equations. An example of what I have in mind is solving $A_{ab} X_{bc} A_{cd} (B_{ad} B_{bc} + B_{ac} ...
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22 views

Quotients in exterior products

I just started learning exterior products. The way I understand it, one can associate a subspace with with a bunch of spanning vectors using an alternating multilinear form. The 'k-blade' remains ...
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1answer
30 views

2-form corresponding to a contravariant vector and pseudo-forms

In Frankel's book he writes that in $R^{3}$ with cartesian coordinates, you can always associate to a vector $\vec{v}$ a 1-form $v^{1}dx^{1}+v^{2}dx^{2}+v^{3}dx^{3}$ and a two form $v^{1}dx^{2}\wedge ...
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1answer
76 views

Do these two rearranged matrices have the same singular values (or the same rank)?

This is the origin of my problem: I have a set of data which expresses which user ($U$ set) applies what tag ($T$ set) to which item ($I$ set). So it is actually a $U×I×T$ tensor $A$ (or 3-dimensional ...
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46 views

How do you solve a linear transformation with no transformation matrix given?

I am stuck, I can't see how Tff was found with no transformation matrix. And now am being asked to find Tgg, help me http://oi60.tinypic.com/33yrplv.jpg
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1answer
42 views

On isomorphisms of tensors of certain type

I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types. Let $V$ be a $k$-vector space. For a field extension ...
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1answer
74 views

Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
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1answer
46 views

Simplying linear equation to get quartic in q using Maple and then using Descarte’s rule of sign

Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots. \begin{equation} ...
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32 views

Is there a definition of the Pfaffian parallel to the definition of the determinant in terms of the exterior algebra?

If $V$ is a finite-dimensional vector space, we have some $A \in \operatorname{End}(V)$ then we can extend $A$ to act on the exterior algebra $\bigwedge(A)$ by setting $$A \cdot (v_1 \wedge \cdots ...
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40 views

Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
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Dimension of the space of tensors obtained by making partial symmetrizations and skew-symmetrizations.

Let $A=(a_{i_1\dots i_k})_{i_1,\dots,i_k=1}^n$ be a higher order cubic tensor or hypermatrix. The following two facts are well-known and are easy to prove: ${(\bf 1) }$ The dimension of the ...
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69 views

Linear algebra puzzle

I stumbled upon an interesting problem in my calculations, which I can't figure out how to solve directly. It goes as follows: show that $$ -2\left(I\otimes (Z-Q)C^{-1} + (Z-Q)C^{-1} \otimes I ...
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1answer
59 views

Tensor product and direct sums

I have an integral domain $R$, and $R$-modules $M$, $N_1$, $N_2$. I know that there is an $R$- module isomorphism $$M\otimes_R (N_1\oplus N_2)\cong (M\otimes_R N_1)\oplus(M\otimes_R N_2).$$ where ...
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1answer
58 views

“Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions

Is there some precise sense in which the "alternation" functor $A$ that maps a multilinear function $f\colon M^d\to N$ to the alternating multilinear function $A(f)\colon M^d\to N$ defined by $$ ...
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2answers
167 views

Exterior power “commutes” with direct sum

I know that for vector spaces $V, W$ over a field $K$, we have the following identity : $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ ...
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4answers
309 views

Tensor product in multilinear algebra

In the book by Halmos ($FDVS$) the tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised ...
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1answer
78 views

Trace of the exterior power as a determinant

Let $A$ be a matrix. According to Wikipedia, $$tr(\wedge^k A) = \frac{1}{k!} \det \begin{pmatrix} tr (A) & k-1 & 0 & \cdots \\ tr (A^2) & tr (A) & k-2 & \cdots \\ \cdots & ...
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1answer
69 views

Can someone what this notation means?

I don't understand what does $\phi_I$ mean The proof includes writing $\phi_I$ as a product of $\phi_{i_1}\phi_{i_2},\dots$, but it doesn't explain what the LHS really means
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1answer
57 views

Calculating a cokernel on wedge product

i have a question... maybe it is easy and im only doing some mistake. Given a surjective homomorphism $f\colon \mathbb{Z}^n \rightarrow \mathbb{Z}^m$, then its kernel $K_f$ is isomorphic to ...
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0answers
27 views

prove that ''pullback'' maps forms to forms

Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all ...
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1answer
28 views

A problem on mutlilinear algebra

In Greub's book on multilinear algebra, a problem asked to show $B(E,F;G)$ is isomorphic to $L(E;L(F;G))$ where $B(E,F;G)$ denotes the bilinear mapping from $E*F$ to $G$ and $L(A;B)$ denotes linear ...