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

learn more… | top users | synonyms

1
vote
1answer
348 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
706 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
786 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
131 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
315 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
220 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
107 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
702 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
138 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
100 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
103 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
409 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
838 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
134 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
599 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
68 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
977 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
638 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
350 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
269 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
73 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 ...
1
vote
1answer
38 views

Can $W(V)\hookrightarrow\operatorname{End}_k(R)$ when $\operatorname{char}(k)=0$?

This is a follow up about a case I'd been meaning to ask in a question I asked about a week ago. Suppose $V$ is a vector space of dimension $2n$, and let $W(V)$ be the associated Weyl algebra, which ...
0
votes
2answers
257 views

Levi-Civita Symbol and index manipulation

Above is the question and answer to question (b) (ignore (a))... I don't get where the final implication comes from. Why can we use c as a (covariant) index on the LHS... surely we must use d or ...
5
votes
1answer
410 views

An Expression for the Wedge Product

For the question below, I have the following definitions and concepts in mind: The $k^{th}$ exterior power of a real vector space $V$, denoted $\Lambda^k(V)$ can be realized as the quotient of the ...
14
votes
2answers
898 views

Proving that the coefficients of the characteristic polynomial are the traces of the exterior powers

Let $T$ be an endomorphism of a finite-dimensional vector space $V$. Let $$f(x)=x^n+c_1x^{n-1}+ \dots + c_n$$ be the characteristic polynomial of $T$. It is well known that ...
7
votes
2answers
305 views

Why is the following map an isomorphism between $Cl(V,\omega)$ and $\operatorname{End}(\Lambda(V))$?

Suppose you have a vector space $V$ of dimension $2n$. I know that there exists a basis $x_1,\dots,x_n,y_1,\dots,y_m$ such that $\omega(x_i,x_j)=\omega(y_i,y_j)=0$ and $\omega(x_i,y_j)=\delta_{ij}$, ...
0
votes
2answers
138 views

Question regarding Tensor product

Let $\ T: V\times W \rightarrow \mathbb V\otimes W$ be a map defined as $\ T(v,w) = v\otimes\ w$ where $\ v \in V,w\in W $. Then T is bilinear. Further if $\ (v_1,\ldots,v_n)\ and\;\ ...
3
votes
1answer
99 views

Homomorphism $W(V)\to\operatorname{End}_k(R)$ is not injective when $\operatorname{char}(k)=p>0$?

Suppose $V$ is a vector space of dimension $2n$, and let $W(V)$ be the associated Weyl algebra, which can be viewed as an associative $k$-algebra with generators $x_1,\dots,x_n,y_1,\dots,y_n$ ...
1
vote
1answer
207 views

Exterior Algebra of Self-Direct Sum

Suppose $V$ and $W$ are vector spaces and $\bigwedge V$ and $\bigwedge W$ their exterior algebras. Then it is known that $\bigwedge (V \oplus W) \simeq \bigwedge V \otimes \bigwedge W$. Now my ...
3
votes
1answer
135 views

Subset of differential operators is a finitely generated module?

I was reading about differential operators, and there is a small claim I don't understand. First, let $A$ be a commutative algebra over $k$, a field. We have the recursive definition for the algebra ...
2
votes
2answers
159 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
8
votes
1answer
859 views

Condition for a tensor to be decomposable

Let $V$ be a vector space of dimension 3 with basis $e_1,e_2,e_3$. Let $W$ be a vector space of dimension 2 with basis $f_1,f_2$. Is $e_1\otimes f_1+e_2\otimes f_2$ decomposable? What about ...
2
votes
1answer
266 views

Basis of the symmetric algebra $S(M)$ given $R$-module basis of $M$ using the diamond lemma?

Over the past week, I read this secret blogging seminar post concerning the diamond lemma, which got me to reading about Bergman's paper on the diamond lemma. Now suppose you have a free $A$-module ...
1
vote
1answer
153 views

A system of three nonlinear equations

I have a system of nonlinear equations. Here it is: $$ \frac{s_2 - K_2}{ps_2^{\gamma_1} + (1 - p)s_2^{\gamma_2}} = \frac{K_1 - s_1}{ps_1^{\gamma_1} + (1 - p)s_1^{\gamma_2}} \\ \frac{s_2}{s_2 - K_2} ...
3
votes
1answer
190 views

Tensors as linear combinations of pure tensors.

Let $V$ be an n-dimensional real vector space, consider the space $F(V^p)$ of real functions on the p-fold cartesian product $V^p$ and its subspace $(V^{*})^p$ of multilinear functions (i.e. covariant ...
2
votes
2answers
94 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
11
votes
2answers
270 views

Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?

I'm trying to understand a certain formula for the determinant in a more general setting. Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
11
votes
1answer
569 views

Sub-determinants of an orthogonal matrix

Let $A$ be a matrix in the special orthogonal group, $A \in SO_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $Det(A)=1$, that is, the column vectors of $A$ make a positively-oriented ...
3
votes
0answers
84 views

Is $M\to M^{\vee\vee}$ injective when $M$ is free?

It's a common theorem that when $M$ is a finite-free $R$-module of rank $n$, there is a natural isomorphism $M\cong M^{\vee\vee}$, where $M^\vee$ denotes the dual. So $M^{\vee\vee}$ is also free of ...
3
votes
1answer
110 views

Why is $M_A\otimes_A N\cong M\otimes_R N$?

I've been doing some tensoring, but am having a hard time understanding the following isomorphism. Suppose $A$ is a commutative $R$-algebra, and for any $R$-module $M$, denote by $M_A=A\otimes_R M$. ...
2
votes
1answer
332 views

Why is the the $k$-th derivative a symmetric multilinear map?

I am having trouble understanding, why the $k$-th derivative of a map $F\colon\mathbb R^n \to\mathbb R^m$ is a symmetric multilinear map for each $x$ in $\mathbb R^n$. Can you please explain which ...
0
votes
1answer
1k views

Hermitian matrices that commute

My question is: If $A$ and $B$ are two Hermitian matrices, and $AB$ is also a Hermitian matrix, then how do prove that both $A$ and $B$ are diagonalizable through the same unitary matrix (i.e the ...
2
votes
1answer
131 views

Symmetric power and characters

Let $V$ be a 2 dimensional vector space over $\mathbb{C}$. Then $W := Sym^{n}(Sym^{m}V)$ is a representation of $GL(V)$. For $g \in GL(V)$, I consider $\chi_{W}(g)$. Let $x$ and $y$ denote the ...