Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
3
votes
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 ...
1
vote
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:
$$\...
0
votes
1answer
40 views
Why a matrix-by-matrix derivative is actually a tensor?
For matrices $A$ and $B$, I thought $\frac{\partial A}{\partial B}$ is a matrix $C$ where $C_{ij} = \frac{\partial A_{ij}}{\partial B_{ij}}$.
However, when I use this matrix calculus website, it says
...
0
votes
0answers
52 views
Inner product structure on geometric algebra?
I understand that geometric algebra equips itself with the contraction operators $\rfloor$ and $\lfloor$. While these are awesome when one wishes to project a subspace onto another, it is not an inner ...
0
votes
1answer
33 views
A linear transform mapping one billinear map to another may not exist (counter-example)
$$\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
votes
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 \...
1
vote
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 ...
0
votes
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
votes
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
vote
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 ...
1
vote
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
$$
\det\left(\begin{bmatrix}
a_{1}+b_{1}x & a_{1}-b_{1}x & c_{1}\\
a_{2}+b_{2}x & a_{2}-b_{2}x & c_{2}\\
a_{3}+b_{3}x &...
0
votes
1answer
58 views
L1 norm minimization over a matrix for a linear system
Let $\mathbf{A} \in \mathbb{R}^{m \times n}$, where $m<<n$ and $\mathbf{b} \in \mathbb{R}^{m}$. The rank of $\mathbf{A}$ is $m$ and both $\mathbf{A}$ and $\mathbf{b}$ are known. Consider the ...
0
votes
2answers
43 views
How to construct the matrix representation of a bilinear transformation?
Suppose $f\colon \mathbb{R}^m\times\mathbb{R}^m\to\mathbb{R}^n$ is a bilinear transformation. How do I define and construct the matrix representation of $f$ with respect to the canonical bases of the ...
0
votes
0answers
22 views
What operation makes a multi-row matrix from a single-row matrix?
Suppose we have the matrix:
$$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&...
0
votes
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
votes
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
votes
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
votes
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 $\...
1
vote
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
votes
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 ...
2
votes
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 ...
-1
votes
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 ...
0
votes
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 ...
1
vote
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
...
1
vote
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?
...
1
vote
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 <...
0
votes
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 ...
1
vote
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 ...
1
vote
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,...
0
votes
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 ...
0
votes
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
votes
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}...
4
votes
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 ...
0
votes
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 $\bigwedge^k V$....
0
votes
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
votes
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 \...
2
votes
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
votes
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
vote
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
$$u = e_1 \otimes e_1 \otimes e_1 + e_2 \otimes e_2 \otimes e_1 + e_1 \...
0
votes
0answers
26 views
A special construction in $\mathbb{R}^n$
I have a question from my professor' notes. We defined $\Lambda^k(V)$ as the set of all $k$-Tensor' forms (multilinear transformations), $\omega$, which fulfuill $\omega(v_1,...,v_i,v_j,...,v_k)=-\...
1
vote
2answers
93 views
Do complexification and exterior power commute?
Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$.
Are $(\bigwedge^k V)^{\mathbb{C}}$ and $\bigwedge^k (V^{\mathbb{C}})$ naturally isomorphic?
They both have the same complex ...
1
vote
0answers
44 views
Induced inner product on tensor powers.
Let $V$ be a real or complex inner product space with inner product $\left\langle \cdotp,\cdotp\right\rangle$. For $\otimes ^kV$ define $\left\langle \cdotp,\cdotp\right\rangle_k$ by
$$\left\langle ...
0
votes
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 ...
2
votes
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 ...
1
vote
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
$$df(a_1,\dots,a_n)(h_1+\dots+h_n)=f(h_1,a_2,\dots,a_n)+f(a_1,h_2,a_3,\dots,a_n)+f(a_1,\dots,a_{n-1},h_n)\tag{1}$$
...
0
votes
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 \...
1
vote
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
votes
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
vote
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 ...
