Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
1,338
questions
1
vote
0
answers
12
views
Equivalence of the two functors $Alt^{k}(-*)$ and $(Alt^{k})^{*}$
I am finishing Vector Analysis of Klaus Jänich. I am stuck at chapter $12$ because I am confused about a notation. I hope some of you could untangle it for me.
Lemma
We can interpret each $\varphi \in ...
1
vote
1
answer
122
views
Grounding the concept of a Free Vector space of the cartesian product of two vector spaces
$\def\tv{\tilde{v}}$
$\def\tw{\tilde{w}}$
$\def\F{\mathbb{F}}$
In constructing the Tensor Product of two, finite-dimensional, vector spaces $V,W$ over a field $\F$ it is common to start from the Free ...
- 896
0
votes
0
answers
64
views
$(p, q)$ tensors and multidimensional arrays
I am trying to understand connections between different interpretations of tensors. In many contexts, tensors are treated simply as multidimensional arrays.
Let us consider the following example. Let
$...
- 1,484
0
votes
1
answer
109
views
Definition of tensor product seems to contradict universal property
$\def\vc#1{\vec{\mathbf{#1}}}$
$\def\cv#1{\tilde{\mathbf{#1}}}$
$\def\qty#1{\left(#1\right)}$
$\def\F{\mathbb{F}}$
In a number of standard textbooks on tensors for physics students (e.g. Tensors: The ...
- 896
1
vote
0
answers
35
views
A countable tensor product: $\ldots \otimes M_{-1} \otimes M_0 \otimes M_1 \otimes \ldots$
It is well-known that if $R, S, T$ are rings, $A$ is an $(R, S)$-bimodule and $B$ is a $(S, T)$-bimodule, we can form the $(R, T)$-bimodule $A \otimes_S B$ as the quotient of $A \otimes_\mathbb{Z} B$ ...
- 1,649
0
votes
1
answer
56
views
Constructing a basis for a tensor product
Let $V$ and $W$ be vector spaces of a field $\mathbb{K}$. Let $\left\{v_i : i \in \mathcal{I}\right\}$ be a basis of $V$, and $\left\{w_j : j \in \mathcal{J}\right\}$ be a basis of $W$. Let $V \otimes ...
- 1,484
0
votes
0
answers
49
views
Werner Greub's formulation of the Universal Property of the Tensor Product
$\def\id{\operatorname{id}}$
$\def\Im{\operatorname{Im}}
$In Section 1.4 of Multilinear Algebra Werner Greub starts with a bilinear map $\otimes: E \times F \rightarrow T$ where $E,F,T$ are vector ...
- 896
0
votes
0
answers
35
views
Equivalent definitions of tensor power of a vector space
I have two definitions of the tensor power $T^nV$ of a vector space $V\in \bf{kVect}$.
For every $n$-multilinear map $f:V\times ...\times V\rightarrow W$, there exists a unique $\bar{f}:T^nV\...
- 906
1
vote
0
answers
39
views
About Hom (TM ,Hom (TM,ν))≅ Hom (TM⊗TM,ν)
In the book ," Characteristic Classes " by J.W.Milnor and J.D.Stasheff there is a problem ( 5-B) in which the following isomorphism had been mentioned:
Hom ($TM$ ,Hom ($TM ,\nu )) \cong $ ...
- 61
1
vote
1
answer
81
views
Nice proof that $\text{Alt}$ is natural.
There is a pair of functors $T:\text{kVect}\rightarrow \text{kAlg}$ and $\Lambda:\text{kVect}\rightarrow \text{kAlg}^-$ which are left adjoints to the forgetful functors $U$ (forget the multiplication ...
- 906
1
vote
0
answers
32
views
Sign of the permutation when I show that $\ast \ast w= (-1)^{n(n-k)} w$ for the Hodge operator
Let $w=\sum_{I} a_{I}dx_{I}$ be a $k$-form in $\Bbb R ^n$. Let us consider the Hodge operator in a combinatorial form, i.e. as an $(n-k)$ form such that
$$\ast(dx_{i_{1}} \wedge \cdots \wedge dx_{i_{...
- 1,810
0
votes
0
answers
15
views
What if a CP/PARAFAC tensor decomposition can be further decomposed, "recursively"?
Standard CP/PARAFAC decomposition:
A tensor $\mathcal{T}$ in shape $(I_1,\dots,I_N)$ is produced by $N$ matrices $\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}$ where each $\mathbf{A}^{(n)}$ is in shape $(...
- 213
1
vote
0
answers
34
views
Exterior and symmetric powers without choice
Let $R$ be a commutative ring, $F$ a free $R$-Module and $n\in \mathbb{N}$. Can it be proven in ZF that the canonical projections $F^{\otimes n}\twoheadrightarrow \bigwedge^n(F)$ and $F^{\otimes n}\...
- 657
0
votes
1
answer
43
views
Expansion of a p-form in basis one-forms - antisymmetry of coefficients
I have found the following equation for the expansion of a general p-form in a book:
$\phi = \sum_{i_1<i_2<...<i_p}\phi_{i_1,...,i_p}\sum_{P\in S_p}sgn(P) e^{i_{P1}} \otimes e^{i_{P1}} \...
- 3
5
votes
0
answers
59
views
Can we construct the exterior algebra just from simple multivectors?
$
\newcommand\K{\mathbb K}
\newcommand\Ext{\mathop{\textstyle\bigwedge}}
\newcommand\Lip{\mathrm{Lip}}
\newcommand\ev{\mathrm{ev}}
\newcommand\Gr{\mathrm{Gr}}
$Let $V$ be a finite-dimensional $\K$-...
- 7,539