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

learn more… | top users | synonyms

1
vote
1answer
36 views

2-form associated with a skew map

Given a two-form $\omega\in \Lambda^2V$ for some (say finite dimensional) vector space $V$ we may associate with $\omega$ a skew map $f_{\omega}:V\rightarrow V^*$ given by $X\mapsto \iota_X\omega$, ...
10
votes
5answers
2k views

Why is the tensor product constructed in this way?

I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that ...
5
votes
4answers
5k views

Multivariate Taylor Expansion

I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and ...
7
votes
1answer
751 views

Kostrikin's Definition of Tensor Product

I'm having serious trouble to understand the definition of tensor products from Kostrikin's Linear Algebra and Geometry. Until now I've understood a tensor as a multilinear map from the cartesian ...
2
votes
1answer
65 views

Clarifying Theorem 4.11 of Lang's Algebra textbook.

Can someone more explicitly describe Theorem 4.11 in Algebra? Let $E$ be a module over a commutative ring $R$, and let $v_1,\dots,v_n$ be elements of $E$. Let $A=(a_{ij})$ be a matrix in $R$, and ...
0
votes
0answers
41 views

The simplicity of $\bigwedge^i \mathbb{C}^{n+1}$ as a representation of $\mathfrak{sl_{n+1}}$ and its weight vectors

I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors. So ...
7
votes
1answer
1k views

tensor product with dual space

I will explain what I know, and then I will ask my question. Let $V$ and $W$ be vector spaces such that at least one is finite dimensional. In class, we showed that if either $V$ or $W$ is finite ...
2
votes
1answer
46 views

Homogenous Polynomial Functions and the Symbol of a Differential Operator

I have a trivial question concerning Lawson/Michelsohn's "Spin Geometry", Chapter III.§1. There, the symbol of a differential operator $P$ is defined to be a section $\sigma(P)$ in the bundle ...
5
votes
1answer
271 views

Multivariate Gaussian equivalent for a Gaussian integration identity.

For a one-dimensional x, $$\int_{-\infty}^{\infty}x^{2}e^{-x^{2}}dx=\frac{1}{2}\int_{-\infty}^{\infty}e^{-x^{2}}dx$$ This can be shown through integration by parts. There is a good derivation of ...
1
vote
1answer
49 views

Dimension $V_1\times…\times V_n$

I have n Vector spaces $V_1,...,V_n$ and would like to show that $\dim(V_1\times...\times V_n)=dimV_1+...+dimV_n$ Is it possible to show this relation using somehow that the dimension of the tensor ...
1
vote
1answer
74 views

Bilinear extension of a map defined only on pairs of independent vectors

Let $V={\mathbb R}^d$ and $$ A=\bigg\lbrace (v_1,v_2) \in V \times V \bigg| \ v_1 \ \text{and} \ v_2 \ \text{are linearly independent} \bigg\rbrace $$ Consider the maps $f:A \to {\mathbb R}$ ...
2
votes
3answers
551 views

Any suggestions for abstract algebra-multilinear algebra books?

I want to read a little about these: The characteristic polynomial and minimal polynomial of a $T \in\mathrm{End}(V)$, or given a matrix $A$, finding the Jordan form and when can I say it is ...
-2
votes
1answer
95 views

Topology.Linear transformation

Let $T:V \rightarrow W$ be a linear transformation and $S \in L^k (W).$ Verify that $T^*(S^{\delta})= (T^* (S))^{\delta}, \delta \in S_k.$ Here is what I did, but unfortunately it is wrong. ...
3
votes
0answers
353 views

What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
2
votes
1answer
59 views

Is there a smallest sub-Grassmann algebra containing a given vector in Grassmann algebra?

Let $V$ be a $d$-dimensional $\mathbb C$-vector space and the Grassmann algebra $$\mathcal G (V):=\bigoplus_{n=1}^d V^{\wedge n}$$ where $\wedge$ denotes the antisymmetric tensor product. I was ...
1
vote
0answers
30 views

Relation between symmetric powers and G-linear morphisms

My multilinear algebra is pretty bad, so I just wanted to check if I my intuition is correct: $$ Hom_{S_n}(V^{\otimes n}, V^{\otimes n}) \cong Sym^n(End(V)) $$ where V is a finite dimensional vector ...
1
vote
1answer
120 views

A basis for k-tensors

Let $V$ be a vector space of dimension $n$ with basis $e_1, \dots, e_n$. Let $a_1, \dots, a_n$ be the dual basis for $V^\ast$. Show that a basis for the space $L_k(V)$ of $k$-linear functions on $V$ ...
2
votes
2answers
175 views

Showing that a function is a tensor

I am trying to solve question 4 in Munkres Analysis on Manifolds section 26. The question is determine if the following is a tensor on $\mathbb{R}^4$ and express those that are in terms of the ...
1
vote
0answers
83 views

Two questions about generalizing multilinear maps.

A multilinear map from the product $V_1\times\ldots\times V_n$ of vector spaces over the same field $K$ to another vector space $W$ over $K$ is a map $\phi$ such that if we fix vectors $$v_1\in ...
1
vote
0answers
44 views

Ring of invariants for the action of rotation groups in tensors.

Consider the component-wise action of the group $SO(p)\times SO(q)$ in the tensor product of two real vector spaces $S^2(R^p)\otimes R^q$. How to parametrize orbits of this action ? For $q=1$ we ...
3
votes
1answer
44 views

What is the rank of this linear map defined on big and abstract spaces.

Let $V$ be a finite-dimensional space, and let ${\cal L}(V)$ denote the space of all endomorphisms of $V$. For any $\phi \in {\cal L}({\cal L}(V))$, there is a unique bilinear map ${\cal L}(V) ...
3
votes
1answer
79 views

Describe the invariant bilinear maps on the linear group

Apologies if this is a stupid question ; it is at least a natural question. Let $V$ be a finite dimensional space over $\mathbb R$ or $\mathbb C$. Denote by ${\mathcal L}(V)$ the vector space of all ...
1
vote
1answer
354 views

Geometric meaning of Gram determinant

Let $v_1,v_2$ be vectors in $\mathbb{R}^4$. Let $M$ be the $2\times 4$ matrix with rows $v_1,v_2$ in this order. The Gram determinant of $M$ is defined as the determinant of the $2\times 2$ matrix ...
5
votes
2answers
726 views

Exterior power of a tensor product

Given 2 vector bundles $E$ and $F$ of ranks $r_1, r_2$, we can define $k$'th exterior power $\wedge^k (E \otimes F)$. Is there some simple way to decompose this into tensor products of various ...
8
votes
3answers
802 views

The determinant function is the only one satisfying the conditions

How can I prove that the determinant function satisfying the following properties is unique: $\det(I)=1$ where $I$ is identity matrix, the function $\det(A)$ is linear in the rows of the matrix and ...
5
votes
0answers
134 views

Exterior algebra of a subspace

Let $E$ and $E^\star$ be two vector spaces in duality according to a (possibly symmetric) non-degenerate bilinear form $\langle\cdot,\cdot\rangle:E^\star\times E\to\mathbb{R}$. Let $F$ be a subspace ...
2
votes
1answer
321 views

A generalization of Lagrange identity

Let $k,n$ be positive integers, $k\le n$. Let $v_1,\cdots,v_k$ be vectors in $\mathbb{R}^n$. Let $M$ be the $k\times n$ matrix with rows $v_1,\cdots,v_k$ in this order. The Gram determinant of $M$ is ...
5
votes
1answer
222 views

Methods of Multilinear Algebra in Representation Theory

I have been interested in representation theory lately in particular on that of Lie algebras. Now I have noticed that one way of building representations is to take tensor/exterior/symmetric powers. I ...
2
votes
1answer
110 views

3-dimensional array

I apologize if my question is ill posed as I am trying to grasp this material and poor choice of tagging such question. At the moment, I am taking an independent studies math class at my school. This ...
7
votes
2answers
713 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
1
vote
1answer
139 views

What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$?

Let $V$ denote an $n$-dimensional real vector space, and $\wedge^n$ denote the $n$-fold exterior product. What is the isomorphism between $\wedge^n(V)$ and $\mathbb{R}$? In the book Introduction to ...
5
votes
1answer
102 views

General trace relation

Let $V$ be vector space $\dim V=N$, and $A\in End(V)$. Denote $$ \wedge^k A^m(\mathbf{v}_1\wedge\dots\wedge\mathbf{v}_k)=\sum_{s_1,\dots,s_k=0,1,\sum_j s_j=m} A^{s_1}\mathbf{v}_1\wedge\dots\wedge ...
4
votes
1answer
104 views

Trace of the multiplication operator

Let $V$ be vector space, $\dim V=N$. Define the multiplication operator $L_{\mathbf{b}}$ as $L_{\mathbf{b}}:\omega\to \mathbf{b}\wedge\omega$, where $\omega\in\wedge V$ ($\wedge V$ is the entire ...
3
votes
1answer
415 views

Laplace expansion

This statement is from the book of Winitzki Linear Algebra via Exterior Products. (Section 3.4, page 123) Let $V$ be finite dimensional vector space, $\dim(V)=N$. The determinant of the matrix ...
3
votes
2answers
858 views

Abstract linear algebra, trilinear forms

Let $V$ be an 3-dimensional vector space over $\mathbb{R}$. Let $\Lambda^3V^*$ denote the space of alternating trilinear forms on $V$. Note: An alternating trilinear form on $V$ is a map $\omega: V ...
1
vote
2answers
136 views

The Vector Space over another Vector Space

Is it possible to consider a vector space over another vectorspace instead over a field as usual, where came into play that we need a field? And in such a vector space, the vector could be represented ...
7
votes
1answer
613 views

how to understand the tensor product canonical line bundle $\otimes$ dual bundle

Suppose we have a Riemann surface $M$ together with a holomorphic vector bundle $E \to M$ of rank n. let $K$ denote the canonical line bundle and let $E^*$ denote the dual bundle I am trying to ...
2
votes
1answer
95 views

Computing the Hodge-* for a Scalar

Let $(E,g)$ be a real oriented inner product space with orthonormal basis $(e_1, \dots, e_n)$ with corresponding dual basis $(e^1, \dots, e^n)$. Then, for any $\beta \in \Lambda^0(V) := \mathbb{R}$, ...
1
vote
0answers
69 views

Step in Proof of Existence of Hodge-*

I am following the proof of the existence of the Hodge-* operator in Naber's Geometry, Topology and Guage Fields. Given a basis $(e_1, \dots, e_n)$ for a vector space and it dual $(e^1, \dots, e^n)$ ...
7
votes
3answers
1k views

Tensors: Acting on Vectors vs Multilinear Maps

I have the feeling like there are two very different definitions for what a tensor product is. I was reading Spivak and some other calculus-like texts, where the tensor product is defined as $(S ...
1
vote
1answer
77 views

Graded Vector Spaces and the Interchange Law

I'm a little confused about how to correctly interchange factors in tensor products on graded vector spaces. In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded vector ...
5
votes
2answers
664 views

Multilinear optimization

Are there any efficient algorithms to solve, multi-linear objective and multi-linear constraint optimization problems? The multilinear functions are sums of bilinear, trilinear (and so on) terms ...
5
votes
1answer
358 views

Kernel of the Tensor Product of a Linear Map with Itself

For two vector spaces, $V$ and $W$, and a map $f: V \to W$, it is clear that: $$ \ker(f) \otimes V + V \otimes \ker(f) \subseteq \ker(f \otimes f). $$ Does the opposite inclusion hold? If so, I'd ...
3
votes
1answer
97 views

Linear independence regarding Exterior Power .

I have been trying to learn the proof of dimension of exterior power from this text : http://www.thehcmr.org/issue1_2/poincare_lemma.pdf.( Page 16) I am not able to understand the part of linear ...
2
votes
1answer
39 views

is there a way to bound the following 2-norm?

Let $C$ be a three-dimensional tensor of dimensions $n\times n\times n$. Define: $$[C(x,y)]_k=\sum_{i,j}C_{ijk}x_iy_j,$$ i.e. $C(x,y)$ is a vector of dimension $n$. Is there a way to bound the norm: ...
2
votes
1answer
272 views

Tensor Algebra of Tensor Algebra

Suppose $V$ is a vector space and $T(V)$ the tensor algebra of $V$. What happens if we take $T(T(V))$ that is the tensor algebra of the (vector space) $T(V)$? I 'guess' I heard that $T(T(V)) \simeq ...
2
votes
2answers
64 views

is there a tensor that does the following?

I want a tensor (in the multi-linear algebra sense) which takes as an input a matrix $A$ of size $n \times n$ and returns as output an $n \times n$ matrix which is diagonal (zero off-diagonal), and on ...
4
votes
2answers
145 views

Proof that the $d$-th powers generate the $d$-th symmetric power of a vector space

Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ ...
0
votes
1answer
76 views

Tensor Product Question

For a finite dimensional vector space $V$, is it true that $\bigwedge^{n - 1}V \otimes V = \bigwedge^{n}V \oplus \ker(\bigwedge^{n - 1}V \otimes V \overset{\psi}{\rightarrow}\bigwedge^{n}V)$ where ...
3
votes
0answers
172 views

How to invert this function on matrices which involves the permanent?

I'm interested in understanding whether a particular natural function on matrices, closely related to the permanent of a matrix, is invertible, and whether its inverse admits a simple closed form. The ...