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Questions tagged [multilinear-algebra]

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

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1answer
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Intuition behind existence of mixed volumes?

Consider "Volume" as a function from set of $d$-dimensional convex bodies to real numbers. This function is homogeneous of degree $d$ (under rescalings of the convex bodies). Minkowski's theorem ...
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1answer
17 views

$\mathbb{P}_{n,m} \sim \mathbb{P}_n \otimes \mathbb{P}_m$…

let $\mathbb{P}_{n,m}$ be a set of polynomials $P(x,s)$ with complex coefficients such that $P(x,s) = 0$ or $deg(P(x,1)) \leq n-1 $ and $deg(P(1,s)) \leq m-1$ show that $\phi: \mathbb{P}_n \otimes \...
2
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1answer
48 views

Finding the inverse of a map from $\wedge^{k}\Bbb V^*\to \text{Hom}(\wedge^{n-k}\Bbb V,\wedge^n\Bbb V^*)$.

I am new to differential geometry and I have encountered a problem regarding $k$-forms and multilinear algebra. Let $\Bbb V$ be a vector space of dimension $n$ and let $0\leq k\leq n$. For any $\...
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2answers
64 views

If $V=k[x]$ then $V \otimes_k V \simeq k[x,y]$

Let $V=k[x]$. Show that $V \otimes_k V \simeq k[x,y]$. I consider the function $\phi : V \otimes_k V \to k[x,y]$ given by $\phi(f(x) \otimes g(y)) = f(x)g(y)$. I could show that $\phi$ is injective, ...
3
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0answers
29 views

How to write an undergraduate level paper on topics of multilinear algebra or tensor?

I'm a undergraduate student majoring in Mathematics. My professor only suggested me considering the topics in multilinear algebra or tensor, but without any other specific instruction. This topic is ...
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0answers
38 views

Why are exterior products so much “wigglier” than symmetric and tensor products?

Apologies in advance that this is a somewhat soft question. Let $k$ be an infinite field. Fix a dimension $d$ and let $v_1,\dots,v_r$, $w_1,\dots,w_r$ be two tuples of linearly independent vectors in ...
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1answer
50 views

What is the geometric interpretation of a multilinear subspace?

If we have two linearly independent vectors in n-dimensions, we know that they span a plane, for example. In general, they form a subspace. I got introduced to multilinear algebra somehow, but I ...
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2answers
73 views

Mechanics of a $(0,2)$-tensor and a bi-vector acting on 2 vectors

Both bivectors and $(0,2)$-tensors are mathematical structures that take in $2$ vectors and produce a scalar. Similar as in this prior post I wrote, I would like to dumb down the mechanics of these ...
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0answers
25 views

Skew-symmetric implies alternating for $2$ a zero divisor in $R$?

In Keith Conrad's notes: https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf Theorem 2.10 reads: Let $k\geq 2$. If $2\in R^\times$, then a multilinear function $f:M^k \to N$ which is ...
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1answer
42 views

Representing a multi-affine map as a determinant

Given a multi-affine map $f: \mathbb{R}^n \mapsto \mathbb{R}$, is it always possible to represent the function as a determinant? And is there a principled way to generate the matrix if it is possible? ...
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2answers
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Lemma 4.3. Aluffi Algebra Chapter VIII.

The following is from the book Aluffi's “Algebra. Chapter 0” : How the (red-circled) equality holds? Esp. the l.h.s of the equality is $λ_{i_1 \dotsi_l}$ for one chosen ordered $i_1 < \dots <...
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1answer
32 views

wedge product (exterior algebra)

I got confused on the operator of the wedge product on other 2 vectors. Please help. Let $V=\mathbb R^3,e_1= (1,0,0),e_2= (0,1,0)$, and $e_3= (0,0,1)$. Find: $3e_1∧4e_3((1,α,0),(0,β,1))$, where α,β ...
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1answer
42 views

Weighted inner product with arbitrary matrix?

An inner product can be written in Hermitian form $$ \langle x,y \rangle = y^*Mx $$ that requires $M$ to be a Hermitian positive definite matrix. I have read that using Hermitian positive definite ...
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1answer
16 views

Using a Euclidean norm to bound a $k$-tuple

This does not look too complicated, but I've been stuck here for several hours. My question is to prove that $||(h, \cdots, h)||\leq ||h||^{k}$, where $||\cdot||$ is the euclidean norm, and $(h,\cdots,...
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Which subspaces of exterior power have decomposable bases?

Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder: Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
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38 views

$G$ is Lie group and $V$ is a representations of $G$,prove representations $V \otimes V \cong S^2(V) \oplus \Lambda^2(V)$

Let $G$ a Lie group and let $V$ a representations of $G$. Then we have the following representations are isomorphic: \begin{align} V \otimes V \cong S^2(V) \oplus \Lambda^2(V) \end{align} I have no ...
2
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1answer
33 views

Connection between ranks of an endomorphism and its linear image on the exterior power

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$. Let $\psi:\text{End}(V) \to \text{End}(\bigwedge^kV)$ be the exterior power map, $\psi(A)=\bigwedge^k A$. For $B \in \text{End}...
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1answer
62 views

Can we “mod out” a common subspace in the Grassmannian inside the exterior algebra?

While reading this paper, I have seen the following claim stated without a proof: Let $V$ be an $n$-dimensional vector space over a field, and let $\alpha,\beta \in \bigwedge^k V$ be decomposable ...
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1answer
21 views

Is “being decomposable” preserved under taking a subspace?

Let $V$ be a vector space over some field, and $W \le V$ a vector subspace. Let $1<k<\dim V$ be an integer. Suppose $\omega \in \bigwedge^k W$ is decomposable as an element in $\bigwedge^k V$....
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0answers
19 views

What manifold structure is natural for the codomain of a differential form defined on a manifold?

It is well known that a differential form $\omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $p\in M^m $ associates an alternating $r$-linear form $\omega(p)\in ...
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1answer
52 views

Isomorphism between tensor product of vector fields and their dual.

Consider two finite dimensional vector spaces $V_1,V_2$ and their duals denoted by $V_1^{*},V_{2}^{*}$. I am working on a problem that is asking me to prove a generalized version of the below, but I ...
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0answers
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If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ is a complex power, is it a real power up to a sign?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$. Does there exist $M \in \...
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0answers
48 views

Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This question was inspired by this beautiful answer: Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ ($1 \le i_1 < \ldots < i_2 \le 4$) be a basis for $\bigwedge^2V$, ...
3
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1answer
50 views

Is every decomposable basis for $\bigwedge^kV$ “standard”?

This is a curiosity: Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set Let $\omega^{i_1,\ldots,i_k}$ be a basis for $\bigwedge^kV$, whose elements are all decomposable. Is $\...
1
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1answer
22 views

Finding non-singular transformation mapping one tensor to other in $(\Bbb F_2)^{\otimes 3}$

Let $u, v \in V\doteq \mathbb{F}_2^{2 \times 2 \times 2}= \mathbb{F}_2 \otimes \mathbb{F}_2 \otimes \mathbb{F}_2$ be given by $$u = e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_1 + e_1 \...
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0answers
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A special construction in $\mathbb{R}^n$

I have a question from my professor' notes. We defined $\Lambda^k(V)$ as the set of all $k$-Tensor' forms (multilinear transformations), $\omega$, which fulfuill $\omega(v_1,...,v_i,v_j,...,v_k)=-\...
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2answers
54 views

Do complexification and exterior power commute?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Are $(\bigwedge^k V)^{\mathbb{C}}$ and $\bigwedge^k (V^{\mathbb{C}})$ naturally isomorphic? They both have the same complex ...
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0answers
31 views

Induced inner product on tensor powers.

Let $V$ be a real or complex inner product space with inner product $\left\langle \cdotp,\cdotp\right\rangle$. For $\otimes ^kV$ define $\left\langle \cdotp,\cdotp\right\rangle_k$ by $$\left\langle ...
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0answers
24 views

Integrating with respect to a tensor

Without loss of generality, Let's say we have a tensor $A$ of order 3 and a differential form $dV$ of order 2. Then how would we evaluate the following indefinite integral? $$I=\int A dV$$. My first ...
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Why are $(1,0)$ and $(0,1)$ tensors antisymmetric?

The book I'm reading (Nadir Jeevanjee (auth.)-An Introduction to Tensors and Group Theory for Physicists-Birkhäuser Basel (2015)) defined an antisymmetric tensor of type $(r,0)$ or $(0,r)$ as "one ...
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1answer
88 views

Derivative of cross product

Let $f:V_1\times\dots\times V_N \to W$ be multilinear. Then $f$ s differentiable and $$df(a_1,\dots,a_n)(h_1+\dots+h_n)=f(h_1,a_2,\dots,a_n)+f(a_1,h_2,a_3,\dots,a_n)+f(a_1,\dots,a_{n-1},h_n)\tag{1}$$ ...
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1answer
51 views

How to prove that the definition of exterior product of differential forms is not ambiguous?

In page 91 of book A Visual Introduction to Differential Forms and Calculus on Manifolds the exterior product of two differential forms $\alpha \in \bigwedge^{r}(\mathbb{R}^n)$ and $\beta \in \...
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0answers
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How to determine an integral of a differential form?

Let $$ \eta = x^2 \, dy \wedge dz + yx\,dz \wedge dx + z^3 \, dx \wedge dy $$ Can you show me how to calculate : $$ \int_{\Phi} \eta, $$ where $ \Phi $ is supposed to be the parametrization of ...
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0answers
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Relationship between anisotropic and negative/positive definite

Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false: $(V,<->)$ is ...
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60 views

Determining values of multilinear forms/ differential forms

I started getting into the topic of multilinear forms and differential forms. I find it quite hard to get into. if I solve $$ \Phi ( \begin{pmatrix} 1 \\ 2\\3 \end{pmatrix} , \begin{pmatrix} 4 \\5\\6 ...
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1answer
17 views

Convert from one tensor canonical form to another

Suppose we have two canonical forms $A, B \in \mathbb{F}_2^{2 \times 2 \times 2}$ of a 3-dimensional tensor product space over the finite field with two elements, where $A = e_1 \otimes e_2 \otimes ...
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1answer
37 views

Prove that the image of the bilinear application is not a vector space

Prove that the image of the bilinear application $$ \mathbb{R}^2\times\mathbb{R}^2 \ni \big((x_1,x_2),(y_1,y_2)\big) \mapsto \varphi \big((x_1,x_2),(y_1,y_2)\big) = (x_1y_1,x_1y_2,x_2y_1,x_2y_2)\...
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0answers
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Switch to another object of same tensor 'rank' to introduce a different class of multilinear maps

My idea is very simple: we know that 'tensors' are a class of multilinear maps or object that "maps" in a (multi-)linear manner geometric vectors, scalars. a multilinear map is a function of ...
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1answer
17 views

Multilinear form is alternating if the underlying field has characteristics different from 2

Let $F$ be a field, $V$ be a $F$-vector space of dimension $n$ and $L^k(V,F)$ be the space of $k$-multilinear forms $f:V^k\rightarrow F$. I was reading Henri Cartan's Differential Form, where he ...
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0answers
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Is $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]\rightarrow$ $\mathbb{F}_2$ multilinear?

$t$ : $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]$ x $\mathbb{F}_2[X]\rightarrow$ $\mathbb{F}_2$ $t(f,g,h):=f(0)g(0)h(1)+f(0)g(1)h(0)+f(1)g(0)h(0)$ How can I know if this is multilinear? Do I have to ...
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0answers
26 views

Inverse function being onto.

Follow up on this question: As you can see from this link $\Phi=f^{-1}\circ h$ is well defined and maps $[c,d]$ onto $[a,b]$ now in my textbook the next theorem following what I posted in the link, ...
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0answers
33 views

Expressing tensor product in terms of symmetric product and exterior product

Let $V$ be a $k$-vector space. Is there a way to express $T_n(V)$ (n-fold tensor of $V$) in terms of symmetric products $Sym_i(V)$ and exterior products $\Lambda^j(V)$ ? Maybe this is not a very well-...
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0answers
41 views

The Universal Mapping Property of a free vector space.

Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:X\to V$ satisfying the following universal mapping property. I ...
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5answers
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If there are $2$ linearly independent vectors $x,y \in X$ such that $||x+y||=||x||+||y||$, then the unit sphere $S(X)$ contains an interval

Let $S(X)= \{x \in X: ||x||=1\}$ be the unit sphere in $X$. Assume that there are $x,y\in X$ linearly independent such that $||x+y||=||x||+||y||$. Prove that $S(X)$ contains the following set:$[x,y]=\{...
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0answers
67 views

Are antilinear forms part of the tensor algebra of a $\mathbb{C}$-vector field?

Let $V$ be a finite-dimensional vector space over some field $K$, $V^*$ be its dual and $\mathcal{T}(V)$ be its tensor algebra. If $K$ is either $\mathbb{R}$ or $\mathbb{C}$, then every multilinear ...
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0answers
51 views

How to find the value of this determinant?

I'm wondering how to find the value of this determinant. $$\left[ {\begin{array}{*{20}{c}} 0&{{x_1}}&{{x_2}}&{{x_3}}& \ddots &{{x_n}} \\ {{x_1}}&0&{{x_1}}&{{x_2}}&...
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0answers
29 views

multilinear rank of a tensor

let $A$ be a tensor in $R^{n_1\times...\times n_d}$ if we use the truncated higher order singular value decomposition THOSVD to approximate this tensor to a tensor $B$ of multilinear rank $(s_1,...,...
4
votes
1answer
69 views

Is the length of the multilinear cross product given by this formula? If not, what is the correct formula?

This is an attempt to make this question more specific. We can compute the length of the cross product in $\mathbb{R}^3$ using the formula: $$|x\times y| = \sqrt{|x|^2|y|^2 - |x\cdot y|^2}$$ This ...
2
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0answers
58 views

Musical Isomorphisms

I'm studying from Fecko's Differential Geometry and Lie Groups for Physicists, and in the part introducing metric tensors, Fecko introduces the musical isomorphisms between the tangent and cotangent ...
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0answers
29 views

Tensors, Co- and Contravariance

Recently I have started to learn about Tensor-Calculus and in my head I made some connections/conclusions which I want to know whether I'm right or wrong about. As an example consider a $m\times n$ ...