Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
1,172
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I want to find $m,l,n$ and $k$ and $(A_1,B_1),(A_2,B_2)\in V\times W$ such that $A_1B_1+A_2B_2\neq A_3B_3$ for any $(A_3,B_3)\in V\times W$.
I am reading "Tensor Algebra" by Takeo Yokonuma (in Japanese).
Problem 2 (on p. 329)
Let $V,W$ and $U$ be vector spaces over $k$, where $k$ is a field.
Let $\mathcal{L}(V,W;U)$ be the set ...
- 6,147
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Pullback of a constant coefficient form
Let $\omega=\sum_{I}C_Idx_I$, where $I=(i_1,\dots,i_n)$ is a multi-index and $C_I$ constants, be an $n$-form in $\mathbb{R}^m$, with constant coefficients. Here $dx_I$ means $dx_{i_1}\wedge\dots\wedge ...
- 333
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Tensor Product, proof of isomorphism
Let V and W be two vector spaces and
$$\Phi: Hom_K(\land^rV,W)\rightarrow Alt_K^r(V,W)\\ f \mapsto [(v_1,....,v_r)\mapsto f(v_1\land \cdot \cdot \cdot \land v_r)]$$
I need to show that $\Phi$ is an ...
- 83
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Linear Algebra - Tensor Products
Let $V$ be a finite-dimensional vector space over $\Bbb Q$ and $\bigwedge^kV$ be the $k$-th antisymmetric power. Let $\{ v_1, \dots, v_n \}$ be a basis of V. Define $$\pi: V^{\otimes k} \to \bigwedge^...
- 524
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Proving that $\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$.
I have proved in what follows that
$\ast\ast\alpha=(-1)^{kn+k}\alpha$ for every $k$-form $\alpha$ on $\mathbb{R}^n$ and I would appreciate if someone would check my proof and/or point out how to ...
- 3,102
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Functoriality in Multilinear Algebra
So I just encountered the word functoriality in my script for linear algebra, in the chapter multilinear mappings. I don't really know what it means and after some research it turns out it's a term in ...
- 115
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Solving a matrix equation involving tensors
While doing my research, I came across the following matrix equation in $W \in \mathbb{R}^{n \times d}$ that I could not solve.
$$ \sum_{i=0}^{t} X_{i} W Y_{i} + X'_{i}WY'_{i} = Z $$
where
$X_{i}, X'...
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Computing $\omega^n=\omega\cdots \omega$ ($n$ fold product), where $\omega=\sum\limits_{k=0}^{n}dx_idy_i$
I have proved the following statement and I would like to have some feedback on my proof (is it correct? can it be improved?). Thanks.
Write the coordinates on $\mathbb{R}^{2n}$ as $(x_1,y_1,x_2,y_2,\...
- 3,102
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How to calculate the wedge product of three terms
This question was asked in my assignment on Linear Algebra and I am struck on it because I am not very good in Wedge products .
Question: Suppose the standard co-ordinate on $\mathbb{R}^3$ are x,y,...
- 1,545
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The set of $d\vec{x}_I$ with $I$ increasing is a basis of the vector space $\Lambda^k(\mathbb{R}^n)^*$ of alternating multilinear functions
I am trying to prove that the set of $d\vec{x}_I$ with $I$ increasing is a basis of the vector space $\Lambda^k(\mathbb{R}^n)^*$ of alternating multilinear functions but I am not sure I have done so ...
- 3,102
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Checking my understanding of Einstein summation convention
I have a limited understanding of the convention so I'd like to check. Given a (0,2) tensor or bilinear map $(e_{i}\otimes e_{j})=\zeta_{ij}$ with $i,j=1,2,...,n$, and two covectors $\Omega^{1}=\sum^{...
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Why differ the definition of the tensor product in algebraic and analysis books?
In algebraic books, Tensor product is defined using quotient space and in analysis books, tensor product is defined using the bilinear map and linear functional.
Why is it defined in two ways?
Thanks ...
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What is wrong with my reasoning regarding tensor products?
$\def\Rbb{\mathbf{R}}$
Let $F$ be a subfield of the field $K$ and let $V$ be an $n$-dimensional vector space over $F$. Then $K\otimes_FV\cong K^n$.
Considering the case when $F=K=\Rbb$ and $V=\Rbb^1$, ...
- 1,131
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Dimension of the symmetric\alternating k-tensor over an $n$-dimensional vector space.
I want to solve this question:
Suppose $V$ is a vector of dimension $n$ over a field $F$ of characteristic not equal to 2. Calculate dim $Sym^{k}(V)$(the symmetric k tensor ).
I know that $(Sym^k(V))^*...
- 1,163
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How to prove that $B$ is non-degenerate? does the question has typo?
Here is the question I want to prove:
Let $V = M_2(\mathbb{R})$ be the space of all $2\times 2$ matrices over $\mathbb{R}.$ Show that $$B(X,Y) = det(X+Y) - det(X) - det(Y)$$
Where $X,Y \in V,$ is a ...
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K is an infinite field, f is a nonzero polynomial, prove $\exists \alpha_1,\cdots ,\alpha_n \> \in \mathbb{K}\>s.t. f(\alpha_1,\cdots ,\alpha_n)\neq0$
$\mathbb{K}$ is an infinite field,$f(x_1,\cdots,x_n)\in \mathbb{K}[x_1,\cdots,x_n]$ is a is a nonzero polynomial, prove that $\exists \alpha_1,\cdots ,\alpha_n \> \in \mathbb{K}\>s.t. f(\alpha_1,...
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Uniqueness of Wedge Product
In Lee's smooth manifolds it is an exercise to show that the wedge product
$\wedge \colon \Lambda^k(V)\times \Lambda^l(V) \to \Lambda^{k+l}(V)$ is the unique associative, bilinear and anticommutative ...
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What is a top form?
I'm sorry for such a basic question, but I can't seem to find this term defined anywhere. I'm trying to learn analysis on manifolds, and google has not been any help in figuring out what this thing is....
- 306
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Proof that pullback commutes with exterior derivative
I found what looks kind of like a proof that the pullback commutes with the exterior derivative, but it is so simple that I feel as if it is almost certainly wrong. Here is what I have:
Let $\omega = ...
- 306
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Extend tensor mode-$n$ product to multivariate functions
Given a tensor $\mathcal{X}\in\mathbb{R}^{I_1\times I_2\times\cdots I_N}$ and a matrix $U\in\mathbb{R}^{U_n\times I_n}$, we can define the mode-$n$ product $\mathcal{Y}=\mathcal{X}\times_n U$ by
$$
y_{...
- 439
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93
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Calculating simple pullbacks
I am having great difficulty calculating what seem to be very simple pullbacks. Right now, I am trying to integrate the 1-form defined by $\omega = dy$ parameterized by $f:[0,1] \rightarrow \mathbb{R}^...
- 306
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2
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What allows a bilinear form (output: field element) to also work as a linear map (output: vector)?
A bilinear form $B: V × V → K$, when the inputs are 2 vectors, has 1 element of their field as output.
A linear map $L: V → W$, when the input is 1 vector, has 1 vector as output.
I have seen cases ...
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Do all bilinear forms have a signature? [closed]
If yes: starting from any arbitrary bilinear form, what is the algorithm to calculate its signature?
If no: what are the conditions necessary for a bilinear form in order to have a signature? Is there ...
- 15
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Matrix-rank nonincreasing unitary tensor operations
I have a multidimensional array $A_{ijkl}$ $\in\mathbb{C}^{m\times n\times o \times p}$ indexed by four integers $i,j,k,l$. I will call $i$ and $j$ the "left" indices, $j$ and $k$ the "...
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Generalization of "symmetric positive definite" to higher dimensions?
Suppose I have a positive-definite function $f:\mathbb{R}^n\to \mathbb{R}^1$.
Then the Hessian $\nabla^2 f$ is symmetric positive definite and I can write $\nabla^2 f=QQ^T$ for some real-valued matrix ...
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A $3\times3$ matrix is multiplied by an unknown $3\times1$ matrix and if the result is $\mathbf{0}$ how do I compute the unknown matrix?
So my question is basically how to find the numerical values of $x_1$ $x_2$ and $x_3$ in the following equation.
$$
\begin{bmatrix}
-1 & -2 & -3 \\
-2 & 2 & -2 \\
1 & 2 & 3 \\
\...
- 11
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Solution to a linear equation involving a skew-symmetric tensor
Say that $S(\mathbf{x})$ is a skew-symmetric $k+1$-tensor, that is, $S_{i_0,...,i_a,...,i_b,...,i_{k}}(\mathbf{x})=-S_{i_0,...,i_b,...,i_a,...,i_{k}}(\mathbf{x})$ for $a,b=0,...,k$, then find $S(\...
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Northcott Multilinear Algebra Universal Property Proof
Northcott Multilinear Algebra poses a problem. Consider R-modules $M_1, \ldots, M_p$, $M$ and $N$. Consider multilinear mapping
$$
\psi: M_1 \times \ldots \times M_p \rightarrow N
$$
Northcott calls ...
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Why in the union $\bigcup_{p \in U}T_p^*(\mathbb{R}^n)$ all of the sets $T_p^*(\mathbb{R}^n)$ are disjoint?
I am currently reading the An Introduction to Manifolds by Loring W.Tu (2nd edition, pp. 34), and as a novice to differential geometry and topology it is not quite obvious to me why in the union $\...
- 1,379
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Invariant polynomial function on Lie algebras.
Take $L$ a complex simple Lie algebra and $f \in S(L^*)^L$ where $S(L^*)$ is the symmetric algebra of $L^*$ (that could be seen like algebras of polynomial functions on $L$). For every homogeneous $f \...
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Show that $\big\{A(v_{i_1}\otimes \cdots\otimes v_{i_k})\in V^{\wedge k}:i_1<\cdots<i_k \big\} $ is linearly independent
Let $V$ be vector space over $\mathbb{R}$ and $V^{\otimes k}$ be the $k$th tensor power of $V$.
Denote by $S_k$ the set which contains exactly all the permutations of a set with $k$ elements.
Using ...
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The volume element determined by some inner product and orientation
While solving the problem 4-7 in Spivak's "Calculus on Manifolds", I came up with a hypothesis which can be helpful for solving the problem. The following is the problem which I'm trying to ...
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Why does $\det A$ change sign when any $2$ columns of $A$ are interchanged?
I have tried to reason, using multilinear forms, the well-known fact that the determinant of a matrix $[A]$ changes its sign if any two columns of $[A]$ are interchanged. I am not confident if my ...
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A positive semidefinite bilinear form must be degenerate?
If $\beta$ is a positive semidefinite (symmetric bilinear)/Hermitian form on $V$, i.e $\beta(v,v) \geq 0$ for all $v \in V$ with possibility that $\beta(v,v) = 0$ for $v \neq 0$. Then it satisfes the ...
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Continuity of a basis of a family of quadratic forms
Let $p$ be a parameter in a k-sphere $\mathbb{S}^k$ and let $Q^p$ be a family of real quadratic forms over $\mathbb{R}^n$, which is continuous with respect to $p$.
For every fixed $p$, there exists an ...
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On the coherence of the tensor product between vectors.
Note. All vector spaces are of finite dimension.
The question concerns the coherence between the tensor product of vector spaces and the tensor product between vectors. But first let's introduce some ...
user805324
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How do you show that $e_{I}^{*}(e_{J}):=\delta_{I,J}$ is a basis for $(\Lambda_{k}V)^{*}$, $\Lambda_{k}V:= V^{\otimes k}/A$?
In Tu's book "Geometry" there is the following statement on page 171:
I would like to show that $span\{e_{I}^{*} \}=(\Lambda_{k}V)^{*}$. For some reason I get completely confused by all the ...
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Find the kernel of a strange map of quotients of vector space
Let $V$ be a vector space of dimension $n$ and let $W \subset V$ be a $k$-dimensional subspace. Consider the map
$$
\varphi: (V/W) \otimes V \to \wedge^2(V/W)
$$
which sends for $a,b \in V$
$$
(a+W) \...
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I need to solved this linear algebra equation with an unambiguous solution
You know that
$$\left[\begin{array}{rrr}
a & b & b \\
b & c & -b\\
c & d & a
\end{array}\right]
*\left[\begin{array}{rrr}
a \\
1\\
b
\end{array}\...
- 1
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How to prove this Proposition about determinant functions and linear composition?
Let $\Delta$be a determinant function of $N$ dim vector space $V$.
Let $|v\rangle$and $\{|v_k \rangle \}^N_{k = 1} $ in vector space $V$.
Proof the following eqatuion：
$$\sum_{j=1}^N(-1)^{j-1}\Delta(|...
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How to prove the following determinant identity
Prove:
$$
\begin{array}{|cccccccccc|} 1 & 0 & 0 & \cdots & 0 & 1 & 0 & 0 & \cdots & 0 \\ x & x & x & \cdots & x & y & y & y & \cdots ...
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Embedding from exterior product to tensor product space
I came across the following:
For a basis $\phi_1,\dots,\phi_n$ of $V$ there is a natural embedding $V^{\wedge n}\hookrightarrow V^{\otimes n}$ defined as
$$(\phi_1\wedge \cdots \wedge \phi_n)
\...
user185346
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triple cross product (for beginner)
I was reading triple vector product. I saw an expression like this :
$${\displaystyle {\begin{aligned}(\mathbf {u} \times (\mathbf {v} \times \mathbf {w} ))_{x}&=\mathbf {u} _{y}(\mathbf {v} _{x}\...
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Decomposition of Hodge Operator
Given a decomposition of a vector space $V \simeq U \oplus W$. Then as taking the exterior algebra preserves coproducts (it is left adjoint to the forgetful functor from graded-commutative graded ...
- 414
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About the two induced linear maps by a bilinear map
Suppose that $V, W$ are $n$ dimensional vector spaces on a field $K$ and $b:V \times W \to K$ is a bilinear form. And let $\phi$ and $\psi$ be the maps defined by the folloing:
$$\phi : V \to W^* ; v \...
- 49
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3
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Proving that $\dim(\bigwedge^k(V^*)) = \binom{n}{k}$ without constructing an explicit basis
Text:
Discussion:
I find this argument kind of hard to follow because an explicit basis is never constructed; the argument seems kind of indirect.
I'm relatively comfortable with the first sentence ...
- 3,154
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44
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Reconstructing a cubic form
I am given a number of points $n_1,\ldots,n_\ell$ from an integer binary cubic form $ax^3+bx^2y+cxy^2+ey^3$ (without any information on what $x,y$ generated them). Can I reconstruct the form $(a,b,c,e)...
- 30.9k
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Find a minimal subspace $W_{\alpha}$ in $V$ for a given $\alpha \in \bigwedge^{p}(V)$ for which $\alpha \in \bigwedge^{p}(W_{\alpha}).$
Let $V$ be a vector space of dimension $n$, and $1\leq p \leq n$, show that for every $\alpha \in \bigwedge^{p}(V)$ there exists a minimal subspace $W_{\alpha}$ in $V$ such that $\alpha \in \bigwedge^{...
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Existence of a function from a quotient
On "Introduction to Smooth Manifolds" by John M. Lee, at page 309 we're talking about multilinear algebra. There's a proposition on the Characteristic Property of the Tensor Product Space. ...
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Show that $v_1\wedge\dots\wedge v_k = x_1\wedge\dots\wedge x_k \implies \text{span}\{v_1,\dots, v_k\} = \text{span}\{x_1,\dots, x_k\}$
Let $V$ be an $n$-dimensional space and $v_1,\dots, v_k \in V$ are linearly independent. It is clear that if $x_1,\dots, x_k \in V$ have the same span as $v_1\dots v_k \in V$ then there is a scalar $t$...
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