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Questions tagged [multilinear-algebra]

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

3
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2answers
90 views

Why does a space vector $V$ must have finite dimension to be isomorphic to its bidual $V^{**}$?

There are many different ways to define tensors. Actually it seems that the word "tensor" is applicable to many various concepts/objects. In any case, it also seems that when we use the multilinear ...
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1answer
20 views

Inner product on diffential forms independence ortonormal basis

Suppose $\{e_1,...,e_n\}$ is a positive orthonormal basis for the tangent space at a point $p$ in an oriented n-manifold $M$, then define the inner product on $\Omega^k(M)$, for each $k$, by: $$\...
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1answer
40 views

Why a matrix-by-matrix derivative is actually a tensor?

For matrices $A$ and $B$, I thought $\frac{\partial A}{\partial B}$ is a matrix $C$ where $C_{ij} = \frac{\partial A_{ij}}{\partial B_{ij}}$. However, when I use this matrix calculus website, it says ...
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0answers
52 views

Inner product structure on geometric algebra?

I understand that geometric algebra equips itself with the contraction operators $\rfloor$ and $\lfloor$. While these are awesome when one wishes to project a subspace onto another, it is not an inner ...
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1answer
33 views

A linear transform mapping one billinear map to another may not exist (counter-example)

$$\newcommand{\im}{\mathrm{im}\;}\newcommand{\Span}{\mathrm{Span}}\newcommand{\rank}{\mathrm{rank}\;}$$For a bilinear map $\phi : V \times W \to E$ define the first nullspace as $$ N_1(\phi) = \{ v \...
2
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1answer
55 views

Are there trilinear inner products?

Is there such a thing as a "trilinear inner product"? The definition of an inner product is: Let $H$ be a vector space over $\mathbb{K}\in \{\mathbb{R,C}\}$. An inner product is a map $\langle \...
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4answers
49 views

Define dimension without referring to bases

The dimension of a vector space is the common cardinality of all bases. I would like to define it in a way that does not refer to bases. I only care about finite dimensional vector spaces. First, I ...
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0answers
23 views

A concrete example of Schur functor.

Could any one recommend a book for me that contain a concrete example of Shur Functor or give me an example of a Schur functor. Any help will be appreciated. Thanks!
2
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2answers
42 views

Identifying simple tensors.

Let $S$ be a domain. I want to determine whether or not, every element of $\text{Frac(S)}\otimes_S M$ is a simple tensor, where $M$ is any $S$-module. I couldn't produce a tensor that is not pure in ...
1
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1answer
13 views

Basis for that $T: \mathbb{R}^3 \to \mathbb{R}^3$ is in rational canonical form

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T(x,y,z) = (x+y+z, x+y+z, x+y+z)$ Find a basis for $T$ such that your matrix$(A_T)$ is in rational canonical form. I ...
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2answers
64 views

Show that a determinant equals the product of another determinant and a polynomial function without calculating [closed]

Show without calculating the determinant, that $$ \det\left(\begin{bmatrix} a_{1}+b_{1}x & a_{1}-b_{1}x & c_{1}\\ a_{2}+b_{2}x & a_{2}-b_{2}x & c_{2}\\ a_{3}+b_{3}x &...
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1answer
58 views

L1 norm minimization over a matrix for a linear system

Let $\mathbf{A} \in \mathbb{R}^{m \times n}$, where $m<<n$ and $\mathbf{b} \in \mathbb{R}^{m}$. The rank of $\mathbf{A}$ is $m$ and both $\mathbf{A}$ and $\mathbf{b}$ are known. Consider the ...
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2answers
43 views

How to construct the matrix representation of a bilinear transformation?

Suppose $f\colon \mathbb{R}^m\times\mathbb{R}^m\to\mathbb{R}^n$ is a bilinear transformation. How do I define and construct the matrix representation of $f$ with respect to the canonical bases of the ...
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0answers
22 views

What operation makes a multi-row matrix from a single-row matrix?

Suppose we have the matrix: $$m = \begin{bmatrix}-1 \\0 \\1\end{bmatrix}$$ What standard matrix operations on $m$ would give $m_2$ and $m_3$ below? $$ m_2 = \begin{bmatrix}-1&\vert\\0&m \\1&...
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1answer
51 views

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$? [closed]

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$? Where $n\times m :=\{(i,j):0\le n-1,0\le j \le m-1\}$. Since $\underset{n\times ...
2
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1answer
40 views

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 ...
0
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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
56 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
72 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
33 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
43 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
56 views

What is the geometric interpretation of a multilinear subspace? [closed]

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
79 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
31 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
44 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
50 views

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
36 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
86 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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0answers
17 views

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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0answers
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
65 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
22 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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1answer
65 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 ...
2
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0answers
40 views

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 $\bigwedge^2V$, ...
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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 $\...
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1answer
30 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
26 views

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
93 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
44 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
27 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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0answers
40 views

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
90 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
54 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
65 views

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 ...
0
votes
0answers
10 views

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 ...
0
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0answers
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 ...
1
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1answer
23 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 ...