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Questions tagged [multilinear-algebra]

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

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Proof-verification:$im(\partial(f(x))\subset\mathcal{L}^m_{sym}(E,F)$, if $X\subset E$ open, and $f\in C^1(X,\mathcal{L}^m(E,F))$ with symmetric image

I believe to have solved an exercise of one of the Analysis books I am using to freshen up on the topic (H. Amann and J. Escher's "Analysis 2"), but I have not used one of the assumptions of ...
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18 views

Proof-Verification of a multilinear function

Let $f: \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^k \to \mathbb{R}^p$ be a multilinear function. I want to prove that $$\lim_{(h_1,h_2,h_3) \to \textbf{0}} \frac{||f(h_1,h_2,h_3)||}{||(h_1,...
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31 views

Identifying $L(V_1, V_2, \cdots, V_n)$ with $V_1^* \otimes \cdots \otimes V_n^*$ instead of $V_1 \otimes \cdots \otimes V_n$

Let $V_1, \cdots, V_n$ be vector spaces over $k$; let $L(V_1, \cdots, V_n)$ be the vector space of multilinear maps from $V_1 \times \cdots \times V_n$ to $k$. we can identity elements of $L(V_1, \...
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67 views

Counterexample or proof - infinite dimensional vector space.

I have the following: Suppose that $E $ is an infinite dimensional vector space. Then that there exists a dual space $E^*$ such that the natural injection $\varphi : E^* \rightarrow L (E)$ given by $\...
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1answer
44 views

Show that there exists a dual space $E^*$ such that the natural injection $E^* \rightarrow L (E)$ is not surjective.

Suppose that $E $ is an infinite dimensional vector space. Show that there exists a dual space $E^*$ such that the natural injection $i:E^* \rightarrow L (E)$ defined by $i(e^*) = \langle - , e^* \...
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1answer
19 views

Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{*} $ is a projection operator in $ E^{*} $.

Suppose $ \pi: E \rightarrow E $ and $ \pi^{*}: E^{*} \rightarrow E^{*} $ are dual mappings. Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{*} $ is a projection operator in $ ...
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26 views

Multilinear map defined on a module product and the associativity of module products

Let $(E_i)_{i\in I}$ be a family of $\mathbf{Z}$-modules and let $(J_k)_{1\leq k\leq n}$ be a finite partition of $I$. Suppose $$f:\prod_{k=1}^{n}\prod_{i\in J_k} E_i\rightarrow F$$ is a $\mathbf{Z}$-...
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103 views

Bourbaki's construction of generalized tensor product of modules

Let $(G_\lambda)_{\lambda\in L}$ be a family of $\mathbf{Z}$-modules. Let $\phi:\prod_{\lambda\in L}G_\lambda\rightarrow\mathbf{Z}^{(\prod_{\lambda\in L} G_\lambda)},\,x\mapsto e_x:=(\delta_x(x'))_{x'\...
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38 views

Definition of $\mathbf{Z}$-multilinear mapping

Let $(G_{\lambda})_{\lambda\in L}$ be a (not necessarily finite) family of $\mathbf{Z}$-modules, $H$ a $\mathbf{Z}$-module and $u:\prod_{\lambda\in L}G_\lambda\rightarrow H$. If for each $\mu\in L$, $...
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11 views

Any bilinear function over $V\times V^*$ satisfying automorphism invariance is scaled inner product

Let $E, E^*$ be a pair of dual spaces and assume that $\Phi: E^* \times E \to \Gamma$ is a bilinear function such that $$\Phi(\tau^{*-1}x^*, Tx) = \Phi(x^*, x)$$ for every pair of dual automorphisms. ...
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60 views

All sets of $11$ quaternions satisfy some multilinear $11$th degree polynomial equation with fixed real coefficients

Let $R$ be an arbitrary $n$-dimensional associative algebra, over a field $\mathbb F$. Denote by $P_k(\mathbb F)$ the space of formal $k$'th degree associative non-commutative polynomials, with ...
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1answer
28 views

Greub Multilinear Algebra bilinear mappings question

In Greub, Chapter 1, page 1 it says: Suppose $E$, $F$ and $G$ are vector spaces and consider a mapping $$\psi\colon E \times F \rightarrow G.$$ The set $$S = \{\psi (x,y) \in G \mid x \in E, y \in ...
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19 views

Is the set of all multilinear forms on a vector space $V$ over a field $F$ itself a vector space? [closed]

If so, what is vector addition on this space? Is it the usual pointwise addition of functions?
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21 views

Proof of an Identity using Bilinear Maps

I am attempting to prove the following statement: if $\phi$ is a bilinear map which takes $V_1 \times V_2$ to $W$ where $V_1$ and $V_2$ are vector spaces of dimension $l_1$ and $l_2$ respectively and $...
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67 views

Textbooks in which determinant is defined as an alternating multilinear map

I'm interested in this abstract definition of determinant, i.e. determinant is defined as an alternating multilinear map. Could you please suggest me some Linear Algebra textbooks that define ...
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58 views

What is meant by “how an element in the domain is mapped to its image”.

In the following lecture given by Fredric Schuller, he mentions this during the lecture which is on multilinear algebra (know that $P$ is a set of polynomials such that $p$ $\in$ $P$): “consider the ...
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1answer
55 views

Difficulty interpreting high order derivatives in $\mathbb{R}^n$

If $f:U\subseteq\mathbb{R}^m\to \mathbb{R}^n$ is differentiable function then its derivative $$ f':U\to M_{n\times m}(\mathbb{R})\simeq\mathcal{L}(\mathbb{R}^m;\mathbb{R}^n) $$ can be seen, for each $...
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69 views

Understanding the properties of multilinear alternating maps

I recently came across this proposition in my math textbook: Let $D: \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ be a multilinear, alternating map such that $D(a_1,...
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Wedge product of complexifications is complexification of wedge product

$V$ is a real finite dimensional vector space. Denote by $V_{\mathbb{C}}$ its complexification $V\otimes_{\mathbb{R}}\mathbb{C}$. I have already proved that $V_{\mathbb{C}}\otimes_{\mathbb{C}}V_{\...
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Complexification of real linear map and Hermitian metric

Consider $V$ a finite dimensional real vector space with Euclidean metric $\langle|\rangle$ $\langle|\rangle_{\mathbb{C}}$ the induced Hermitian metric on the complexification $V_{\mathbb{C}}=V\...
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1answer
52 views

From vec-trick to matrix-trick for Kronecker products

for the vec-trick of the Kronecker product, we can write $$ \left(\mathbf{B}^{\top} \otimes \mathbf{A}\right) \operatorname{vec}(\mathbf{X})=\operatorname{vec}(\mathbf{A} \mathbf{X} \mathbf{B}). $$ ...
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34 views

Multivariate Linear Function

Linear Definition is as follows $$L(x + y )= L(x) + L(y) $$ $$L(ax) = aL(x)$$ I get confused with the definition for multivariate linear function. Let's say we have a function like below. $$L(x, y)=...
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33 views

How does the “Alternating Operator” distribute in Tensors?

(I'm not sure that I even phrased the question correctly. I will explain more about this below.) Given a k-tensor $T$, we can define an alternating k-tensor $Alt(T)$ in the following way: where $\...
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22 views

Finding angle between two hyperplanes given basis vectors

There is a Euclidean space $\mathbb{R}^n$. In it there live two hyperplanes of dimension $m$ each. Both hyperplanes pass through the origin $\vec{0}$. The hyperplanes are defined by the corresponding ...
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15 views

Linearization of a polynomial

Arora and Barak in their book, CC A modern approach here may have a typo in (8.15) on the page $161$: $\exists_{X_i}p(X_1,...,X_n)=p(X_1,...,X_{i-1},0,X_{i+1},...,x_n)+p(X_1,...,X_{i-1},1,X_{i+1},...,...
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16 views

Equivalence between the product of a skew symmetric matrix and the product of a bivector and a vector

I stumbled upon the following statement on the wikipedia page (https://en.wikipedia.org/wiki/Cross_product#cite_note-lounesto2001-14) about the cross product: The vector cross product also can be ...
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20 views

Moment of inertia tensor and the tensor definition

A tensor is defined as follows; A tensor of type (r, s) on V is a multilinear map T : $(V^*)^r \times V^s \to R$. The set of tensors of type (r, s) on V will be denoted by $T^r_s(V)$. How does this ...
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1answer
30 views

Tensors: How to construct multilinear maps?

A tensor is defined as follows; A tensor of type (r, s) on V is a multilinear map T : $(V^*)^r \times V^s \to R$. The set of tensors of type (r, s) on V will be denoted by $T^r_s(V)$. It takes r ...
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1answer
30 views

Bilinear Maps between Finite-Dimensional Vector Spaces & the Outer Product

Theorem. A function $f : \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \to \mathbb{R}^{n_3}$ is bilinear iff each component of $f(v,w)$ is a linear combination of terms of the form $v_iw_j$, where $v=\...
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30 views

A question on forms

Let $V$ be a vector space of dimension $d<\infty$, and let $v\in V\setminus\{0\}$. Let $f\in \Lambda^p V$ be such that $v\wedge f=0$. I want to show that there exists a $g\in \Lambda^{p-1}V$ such ...
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2answers
56 views

Trying to understand the Isomorphism between $T^1_1(V)$ and $End(V)$

Let $V$ be an n dimensional vector space. Let $T_1^1$ be the set of bilinear functions $F: V^* \times V \rightarrow \mathbb{R}$ and $End(V)$ be all the set of all linear functions $A: V \rightarrow ...
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Is Geometric Algebra/Geometric Calculus all that it's hyped up to be? [closed]

There appears to be a cult following of geometric algebra/geometric calculus (GA/GC) as developed by David Hestenes. Many questions on stack exchange regarding this. I wanted to make this question ...
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23 views

Is there an analog of polarization for skew-symmetric forms?

Polarization works both ways. Not only can you represent any homogeneous polynomial $f$ of degree $d$ as $F(x,...,x)$ for a multilinear form $F(x_1,...,x_d)$ but also conversely, any symmetric ...
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1answer
60 views

Alternating tensors: Difference between $Alt$ and $\frac{1}{k!} Alt$?

From Dummit Foote Chapter 11.5: pages 452 and 453. I believe I was able to reprove these with $Alt$ instead of $\frac{1}{k!} Alt$. Maybe I made a mistake. What's the difference between $Alt$ and $\...
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22 views

linear independence of symmetric tensors

I am reading a paper that incidentally uses a bit of theory of symmetric tensor spaces. I came across the following claim: If we're given linearly independent vectors $x_1, \ldots, x_n \in \mathbb{...
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38 views

Prove that $f(x,y) = - f(y,x)$

I have the following proof statement but I can't prove it Let $ f : V \times V \longrightarrow \mathbb{K} $ an alternating multilinear map. How can I prove that : $f(x,y) = - f(y,x)$ This is what ...
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When are tensor components invariant of the basis?

I am self-studying Bishop and Goldberg's nice book on Tensors and Manifolds, but got stuck on Exercise 2.13.3: If $A$ is a tensor of type $(r,s)$ such that the components of $A$ are the same with ...
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68 views

Question on the differences in the definitions of what a tensor is

Below are the common definitions of tensor. a. "a tensor is a quantity which transforms according to a definite law under the change of the coordinate system". b. "a tensor is a multilinear function ...
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Doubt on a particular commutative diagram using the Tensor Product construction

I've posted two other questions* $[1]$ $[2]$, discussing and asking about the Tensor Product construction, in particular the "canonical construction", which is the one $[4]$: $$ \frac{F(V \times W)}{...
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Doubt on the notation of the generators of the subspace $S \subset F(V \times W)$ in tensor product construction [duplicate]

I understand the particular form of vectors of subspace $S$ of Free vector space $F(V \times W)$; $$ (v+v',w)-(v,w)-(v',w) $$ $$ (v,w+w')-(v,w)-(v,w') $$ $$ a(v,w)-(av,w) $$ $$ a(v,w)-(v,aw) $$ ...
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21 views

Definition of Wedge Product on Exterior Algebra

I'm confused on a detail of the construction of the exterior algebra $\Lambda(V^*)$ from the individual exterior powers $\Lambda^k(V^*)$ of the dual of an $n$-dimensional vector space $V$. We define ...
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68 views

Doubt on the understanding of the role of Quotient Spaces on Tensor Product construction

I'm studyin,g for the first time, the Tensor Product of Vector Spaces. After the answer of a particular question, $[1]$, I think that I grasped the key point of the role of Quotient Vector Space; the ...
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19 views

Different Definitions of the kth Exterior Power

If $V$ is an $n$-dimensional real vector space then the $k$th exterior power $\Lambda^kV$ can be defined in a bunch of different ways and everyone seems to have their own preference. I've been exposed ...
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Intersection of multilinear ideals

Let $K$ be a field. Let $I$ and $J$ two ideals of $K[x_0,x_1,\ldots,x_n,y_0,y_1,\ldots,y_n,z_0,z_1,\ldots,z_n]$ such that $I\subset K[x_0,x_1,\ldots,x_n,y_0,y_1,\ldots,y_n] $ is generated by ...
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17 views

Suppose S is a set of linearly independent vectors in E, and suppose T is a basis of E

Suppose S is a set of linearly independent vectors in E, and suppose T is a basis of E. Prove that there ir a subset of T which, together with S, is again a basis of E.
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26 views

Let $(x_\alpha )_{\alpha \in A}$ be a basis for a vector space E and consider a vector $ a= \sum_\alpha \xi^{\alpha}x_\alpha$ [closed]

Let $(x_\alpha )_{\alpha \in A}$ be a basis for a vector space E and consider a vector $ a= \sum_\alpha \xi^{\alpha}x_\alpha$ Suppose that for some $ \beta \in A$, $\xi^{\beta} \not= 0$. Show that ...
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37 views

Does it make sense to replace scalar with vector in an inner product space?

I obtain a formulation in my study and I found I can rewrite it as a simple and reasonable form but it needs some special operations as follows. Basically, I just replace the scalars with vectors in ...
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111 views

Given math model to model relations using a tensor, how do we put restriciton on tensor so that it can capture symmetric relations

The goal is to model real life relations between stuff and people. Say we have sets $E$, $R$ and functions $h:E \to V$, $t :E \to W$ and $r:R \to U$, where $V,W,U$ are finite dimentional vector spaces ...
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16 views

property of mode_n tensor product

So given a tensor $\chi = \mathcal{T} \times_1A \times_2B \times_3C$ (assuming compatibility of operations), we need to show $\chi_{(1)}=A\mathcal{T}_{(1)}(C\otimes B)^T$ Here $\otimes$ means ...
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10 views

Diagonal root of symmetric multilinear map

Let $F$ be a symmetric multilinear form from $(\mathbb{C}^n)^n$ to $\mathbb{C}$ with real coefficients (these are polynomials). By symmetric I mean that one may permute the arguments $$ \forall \sigma ...

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