Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
1,228
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Symmetric multilinear map and smooth polynomial
Let $V=\mathbb{C}^d$ and $\varphi\colon V\otimes\cdots\otimes V\to\mathbb{C}$ a linear map such that $\varphi(v_1\otimes\cdots\otimes v_n)=\varphi(v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(n)})$ for ...
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2
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What is the kernel of an alternator $\pi:L^k(V) \to L^k(V)$
Define a funtion $$\begin{align*}
\pi:L^k(V) \to L^k(V)
\end{align*}$$ I call $\pi$ is an alternator.
$L^k(V)$ denotes the space of $k$ tensors on vector space $V$ and
$$
\pi(f) = \sum_{\sigma} \...
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Spectral decomposition for higher-rank antisymmertric real tensors
For any arbitrary anti-symmetric matrix $M_{n\times n}$, there always exists orthogonal matrix $O$ that $M = O\Sigma O^T$, with
$$
\Sigma = \left( \bigoplus_{i=1}^m \begin{pmatrix} 0 & 1 \\ -1 &...
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Let $A : \mathbb{R}^n \to \mathbb{R}^n$ be an orthogonal linear map with determinant $1$. Show that $A^* \circ \star = \star \circ A^*$
Let $A : \mathbb{R}^n \to \mathbb{R}^n$ be an orthogonal linear map with determinant $1$. Show that $$A^* \circ \star = \star \circ A^* : \mathrm{Alt}^k(\mathbb{R}^n) \to \mathrm{Alt}^{n-k}(\mathbb{R}^...
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Show that there exists covectors such that $\alpha = \alpha_1 \wedge \dots \wedge \alpha_{n-1}.$
Let $\alpha \in \operatorname{Alt}^{n-1}(\mathbb{R}^n)$. Show that there exists $\alpha_1, \dots, \alpha_{n-1} \in \operatorname{Alt}^1(\mathbb{R}^n)=(\mathbb{R}^n)^*$ such that $$\alpha = \alpha_1 \...
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Prove $\dim {\rm Im}\ T = \dim W$ in the proof of the dimension theorem for the symplectic complement
I am trying to prove the dimension theorem for the symplectic complement and I am missing a small step right at the end:
Let $ (V,\omega) $ be a symplectic vector space and $W$ a linear subspace of $V$...
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2
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61
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Confusion regarding wedge products
While studying smooth manifolds and differential forms I have come across multiple definitions of the wedge product, and I have been having some trouble seeing the equivalence between them.
At the ...
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Can Lawson's proof (that the canonical inclusion into a Clifford algebra is injective) be fixed?
The first proof in Lawson & Michelsohn's Spin Geometry is known to be wrong. The claim, which appears in a paragraph on page 8 (not in an official proposition), is that the projection map $\pi_q|...
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Difficulty understanding Linear maps between vector spaces as $(1,1)$ tensors on the target vector space
In Nakahara's book "Geometry, Topology, and Physics" there is an exercise that asks to show that if $f$ is a Linear Map between vector spaces: $f: V \rightarrow W$ then it is a tensor of ...
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Lemma regarding dual spaces and multilinear algebra
I'm reading about the following lemma
and trying to understand why $\alpha^I(e_J) = 0$ if $I \ne J$, but I don't understand the argument. Is there an alternative way to understand this? If $I=(i_1,...
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Basis of exterior power of a vector space
Let $V$ be a vector space over a field $k$ with a basis $\{e_i\}_{i\in S}$, where $S$ is a linearly ordered set.
Def. n-th exterior power of the space $V$ is a vector subspace $\Lambda^nV\leq\Lambda V$...
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Basis-independent derivation of the fact that the determinant of the composition of two linear maps is the product of the determinant of each
A maximal form $\omega$ on $V$ is defined as an alternating type $(0, n)$ tensor (where $\dim V = n$). From there, the determinant is defined as,
$$ \det \phi \equiv \dfrac{\omega\left(\phi(e_1), \...
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For which numbers of dimensions is the Kulkarni-Nomizu product an injective/surjective map?
(This question only concerns tensors over finite-$n$-dimensional real vector spaces.)
The Kulkarni-Nomizu product is a multilinear map from of pairs of symmetric rank-2 tensors to the space of ...
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Confusion in the proof of linear independence for tensors
If $V$ is finite-dimensional, choose a basis $\{ |b_{1} \rangle, \dots, |b_{n} \rangle \}$. There's a dual basis $\{ f_{1} , \dots, f_{n} \}$ for $\widetilde{V}$, which is the same as $\mathcal{T}^{1}(...
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Can we often only consider homogeneous elements of exterior or tensor algebras because their products preserve homogeneity?
Background: I'm just an innocent physicist with very little formal training in algebra. So I think of things in very naive, concrete (as opposed to abstract), and non-rigorous ways. I probably won't ...
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1
answer
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What does it mean to "identify" a function $f$ with a $0$-form $\omega_0$ and a $3$-form $\omega_3$?
In class we wrote down the following:
We identify a function $f$ in $U \subset \mathbb{R}^3$ with a $0$-form $\omega_0$ and a $3$-form $\omega_3$ and a vector field $u$ with a $1$-form $\omega_1$ and ...
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Derivation of basis and dimension of a multilinear map $\Psi:V^s\to V$
I'm reading linear algebra and came up to the topic of linear map representations as matrices. More specifically for a dim $n$ vector space $V$, the space $Lin(V,V)$ of maps $T:V\to V$ is isomorphic ...
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How can I show by using Hahn Banach separation theorem?
The convex hull of a set of points S in n dimensions is the intersection of all convex sets containing $S$. For $N$ points $p_1, ..., p_N$, the convex hull $C$ is then given by the expression
$$ C = \...
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When is $Hom(X,Y) \otimes Hom(A,B)$ isomorphic to $Hom(X\otimes A, Y \otimes B )$?
We assume that $A$, $B$, $C$, and $D$ are real vector spaces. We write $Hom(X,Y)$ for the vector space of linear mappings from $X$ into $Y$, as usual.
There exists a linear map
$$Hom(X,Y) \otimes Hom(...
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Mujica's "Complex analysis in Banach spaces" exercise 2.M.
I'm trying to prove it in Mujica's book "Complex analysis in Banach spaces" which states the following. I would be grateful if someone could prove it.
Let E and F be Banach Spaces and let $P=...
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Exercise 1.C. in Mujica's book "Complex analysis in Banach spaces"
Let $E$ and $F$ be Banach spaces. Let $(e_1,\ldots,e_n)$ be a basis for E and let $\xi_1,\ldots,\xi_n$ denote the corresponding coordinate functionals. Show that each $A \in L_a (^m E;F)$ can be ...
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Can all rank two tensors be represented as matrices?
I’m curious about wether every Tensor of second order can be represented as a matrix.
The only possibilities are (to my understanding, correct me if i’m wrong):
$$\mathcal{T}^{(2;0)}= \mathbf{V\otimes ...
1
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1
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How would you classify a tensor space equipped with an addition and scalar multiplication?
I have been thinking about vectors, vector spaces, tensors, and tensor spaces. So far, I have surmised that a tensor space is defined to be $(\otimes_{i=1}^{k} V,+)$ where the addition is vector ...
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Can second rank antisymmetric tensors be constructed from vectors using tensor product?
My background is in physics. I am motivated to understand tensor behind "a tensor is something that transforms like a tensor." My understanding of a tensor in that a second rank tensor is an ...
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Associativity property for multiple tensor product
The tensor product has the following properties. Note that all
vector spaces are over the same field .
(Associativity) There exists an isomorphism
$\tau: (V_{1}\otimes....\otimes V_{n})\otimes(W_{1}\...
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Multilinear functions of degree zero
I apologize in advance, but I need help with a rather simple question.
Given $L_k(V)$ is the set of all $k-$linear functions on the vector space $V$ on $\mathbb{R}$, what is $L_0(V)$? Does it make ...
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How to show that this multilinear map is alternating?
Let $V$ be a vector space over $K$ with basis $\{e_1,\ldots,e_n\}$.
I already showed that for a fixed tuple $I = (i_1,\ldots,i_r)$ with $1 \le i_1 < i_2 < \cdots < i_r \le n$, there exists a ...
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2
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Natural orientation of an underlying real vector space
Consider the following argument from a book (included as an image to show exactly what is printed):
There are a number of minor typos here, but I'm mainly concerned with the computation of $\Delta(...
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Basis of exterior product
Let $V$ be a vector space over $K$ with basis $\{e_1,\ldots,e_n\}$.
For a fixed tuple $I = (i_1,\ldots,i_r)$ with $1 \le i_1 < i_2 < \cdots < i_r \le n$, use multilinear extension to ...
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How does a third-order tensor behave algebraically with respect to standard linear algebra operations?
How does a Hessian tensor behave algebraically?
Suppose $H_x$ is the Hessian tensor of a multivariate function $f:\mathbb{R}^n\to\mathbb{R}^m$. This is a third-order tensor and I am told one can think ...
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Classical reference for “vector-valued” Laplace expansion?
In relation to the $2\times 2$ (say) determinant $$\begin{vmatrix}a&b\\c&d\end{vmatrix},$$ consider the “bordered” determinant
$$\begin{vmatrix}
x&a&b\\
x&a&b\\
y&c&d
\...
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Exercise from Greub, Multilinear Algebra on factorization of bilinear maps [duplicate]
I am really stuck on exercise 5 in chapter 1.3 of Greub's book Multilinear Algebra.
The exercise asks me to show that if $\varphi \colon E \times F \to G$ and $\psi \colon E \times F \to H$ are ...
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Is a third-order tensor linear in both arguments?
Let $f:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m$ be smooth and for each $x\in\mathbb{R}^n$ let $H_x(\cdot, \cdot)$ denote the Hessian third-order tensor of $f$. Basically given any two vectors $v, ...
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Term between "orthogonal" and "orthonormal"
If we have a bilinear form $(-, -)$ on a vector space $V$, a basis $\beta$ of $V$ is said to be orthogonal if $(v, w) = 0$ for every $v \neq w \in \beta$, and orthonormal if moreover $(v, w) = 1$ when ...
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How do you invert a tensor in the EFE equation such as the stress energy tensor or the Newtonian tensor?
Motivation:
Tensors are built from vector spaces because a tensor satisfies the axioms of a vector space under the proper equipment of addition and multiplication. How do we define rigorously the ...
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Alternative norm of $k$-linear operator
For a $k$-linear operator $T \colon \mathbb{R}^n \times \cdots \times \mathbb{R}^n \to \mathbb{R}$ its norm can be defined as:
\begin{equation}
\lVert T \rVert
=
\sup_{ \lVert x_i \rVert_{\mathbb{R}^...
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Different definitions of a tensor
Some Linear Algebra textbooks defines tensors as the k-linear map:
\begin{equation}T:V^{\times p}\to\mathbb{F}\end{equation}
While others define it using the dual:
\begin{equation}T:V^{\times p}\times ...
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1
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67
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Trying to understand how tensors transform under a coordinate system change
I'm trying to understand how tensor components transform under a change of coordinate system, but something doesn't look right. Below is the derivation that I wrote out:
We start with a general ...
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2
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1
answer
65
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Generators for the group of linear transformation preserving $x^2 + y^2 - 2z^2$?
When I have a quadratic form, such as $Q = x^2 + y^2 - 2z^2$, how do I find generators for the group that preserves this quadratic form? That group would also be called $SO(Q)$ or $SO(3,Q)$.
If ...
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Definitions - Tensor Products and Tensors
So, I've been trying to get some grasps on Tensors, and now I've come to look at one particular classical definition, that goes around Tensor Products.
So, I think I was able to follow the ...
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1
answer
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Eigenspaces in the symmetric power
Given a finite dimensional vector space $V$ and a linear map $f:V\to V$, we can construct a multilinear map, which we will denote by $F$, on the $n^{th}$ symmetric power of $V$, which I tend to think ...
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Is this how to prove the two-sided ideal is spanned by $v_1\otimes\dots\otimes v_k$ where the $v_1,\dots,v_k$ are linearly dependent?
Let $V$ be any vector space, (I'm not sure about the characteristic of the field) consider its tensor algebra $TV$. The two sided ideal is defined as the subspace $I \subset TV$ generated by all ...
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How can I express this linear algebra sum of outer products in tensor notation?
I am curious about tensors and tensor notation and how it translates to common linear algebra stuff that I already know. For instance, we can express an outer product $AA^\top$ as a sum of outer ...
- 801
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Endomorphisms as derivations
A derivation on tensors is a map $T \mapsto D T$ that preserves the type of the tensor $T$; is linear; commutes with contractions; and satisfies the product rule
$$
D\left(T_1 \otimes T_2\right)=\left(...
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2
answers
151
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Gradient of $f(x) = (Ax - b)^\mathsf{T} (Ax - b)$ [duplicate]
I want to find the gradient of $f(x) = (Ax - b)^\mathsf{T} (Ax - b)$, from product rule, I got:
$$
\mathrm{d}f(x) = (A \mathrm{d}x)^\mathsf{T}(Ax-b)+(Ax-b)^\mathsf{T}(A \mathrm{d}x)
$$
now I know I ...
1
vote
1
answer
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Basic question on tensor product of two vector spaces
This may be trivial question, but I am trying to clarify my understanding of this concept.
In the universal property of tensor product of $k$-vector spaces $V$ and $W$, it is stated that
it is a ...
- 2,520
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23
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Payoff Functions are Multilinear Forms?
I am reading Nash's paper "Equilibrium Points in $n$-Person Games" in which he claims that:
"For mixed strategies, which are probability distributions over the pure strategies, the ...
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35
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Upper bound on a contraction of tensors
Definitions and assumptions
Let $ k, l, m, n \in \mathbb{N^+} $ be positive integers, and let $ \vec{p} \in \mathbb{R}^k $ be a finite probability distribution, i.e. $ \sum_i p_i = 1 $ and $ \forall i ...
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1
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Formalizing the universal property of the tensor product
Introduction (a bit long)
I am very new to category theory and algebra, in general.
I recently saw the formalized concept of a universal morphism, here in WikipediA.
I will repeat the definition with ...
1
vote
1
answer
26
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Naive question about notation for unusual products of entries of matrix with rank-3 tensor
I am trying to do a computation and I'm not very good with actually using tensors for computations.
I have some $(n\times n)$ matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ and some kind of rank-3 tensor $...
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