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

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Let $V,W$ be (finite dimensional) vector spaces over a field $\mathbb{K}$. Construct the tensor product between them as the quotient $\mathcal{F}(V\times W)/R$ where $\mathcal{F}(V\times W)$ is the ...
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While pondering why the Levi-Civita symbol shows up in the commutation relations for Pauli matrices, I found that $\langle A,B,C\rangle=\text{tr}(ABC)$ is an alternating trilinear form on traceless $2\...
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Every once in a while I run into confusions about tensors. This time, I wonder if my understanding of the following claim is correct. Claim. For two vector spaces $V, W$ (not necessarily finite ...
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In an assignment we had, we were asked to find the covariant, contravariant, and mixed components of a second-order tensor $A$, i.e., $A_{ij}, A^{ij}, {A_{i}}^j, {A^{j}}_{i}$. The first thing that ...
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We can consider the definition of a hyperbolic space in the context of symmetric bilinear forms. Let $\varphi \in B_s(V)$. A two-dimensional subspace $W \subseteq V$ is said to be a hyperbolic plane ...
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Given $\varphi \in Hom_{\Bbb F}^2 (V, W, \Bbb F)$, consider the map ${\varphi}D:V \to W^{\ast}$ (from $V$ to the dual of $W$) defined by $${\varphi}D(v)(w)=D{\varphi}(w)(v)=\varphi(v, w).$$ I am ...
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Sending $\bigwedge_i v_i$ to $\sum_{\sigma \in S_n} \text{sgn}(\sigma) \cdot \bigotimes_i v_{\sigma(i)}$ defines a natural map $\alpha: \bigwedge^n V \rightarrow \text{Alt}^n V$. This is easily seen ...
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A friend of mine is currently writing his Bachelor's thesis on the topic of elastic materials. In particular, this involves higher-order derivatives. These are naturally expressed in the language of ...
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Background : For solving large matrix-vector equation systems like $$\mathbf {Ax = b}$$ I have used Krylov subspace methods for a long time and especially Conjugate Gradient. Being an iterative method ...
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I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu. Let $e_1,\dots,e_n$ be a basis for a real vector space $V$. Let $\alpha^1,\dots,\alpha^n$ be the dual basis. $\...
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Here is the context. Munkres claims “these five properties characterize the product $\wedge$ uniquely for finite-dimensional space $V$”. Then, in Step 10, he tries to verify the uniqueness: Below ...
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Step 1: A function that assigns, to each $A\in\mathbf{R}^{n,n}$, a real number denoted $\det A$ , is called a determinant function if it satisfies the following axioms: (i) If $n>1$ and $B$ is the ...
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Let $\mathbf{a}_{1},\cdots,\mathbf{a}_{n}$ be a basis for the vector space $V$. Choose any $i\in{1,\cdots,n}$. There exists a unique linear map $\phi_{i}\colon V\rightarrow\mathbf{R}$ such that $\phi_{...
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For a real positive definite(semidefinite) tensor $\mathcal{A} \in \mathbb{R}^{n \times n \times \dots \times n}$ of order $m$, we can define: The $\textbf{Frobenius norm}$ as $$\|\mathcal{A}\|_F^2 = ...
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I am looking for a source that defines the polarization of the Pfaffian as a symmetric multilinear form $\widetilde{\operatorname{Pf}}(X_1, \ldots, X_n)$ associated to the usual Pfaffian $\...
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Let $\mathbf{a}_{1},\cdots,\mathbf{a}_{n}$ be a basis for the vector space $V$. Choose any $i\in\{1,\cdots,n\}$. There exists a unique linear map $\phi_{i}\colon V\rightarrow\mathbf{R}$ such that $\...
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Reading Hitchin's Projective Geometry, and particularly the chapter on exterior algebra I found the following definition for the wedge product: But I don't understand what this definition really says,...
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I am trying to understand a very brief note written by the great physicist Asher Peres. In this note he discusses the special case of applying Schmidt's theorem to a third order tensor. Schmidt's ...
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It is known that for matrices, under the correct conditions, the eigenvalues of the matrix are continuous when considered as functions of some perturbation. See for example, here . Similarly, under ...
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A trilinear form is a map $\def\F{\mathbb{F}}\gamma: U \times V \times W \to \F$ where $U,V,W$ are finite-dimensional vector spaces defined over a field $\F$. We can induce a map $U \to \mathcal{P}(\...
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Given two affine spaces $A$, $B$ with direction spaces $\partial A$ and $\partial B$, is it possible to define a tensor product of $A\otimes B$? Obviously we can form the tensor product $\partial A \...
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Let $V_1$ and $V_2$ be finite dimensional real inner product vector spaces with $\dim V_2 > \dim V_1$ and let $I: V_1 \rightarrow V_2$ be a linear isometry with adjoint $P$. Consider positive ...
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Background I'm writing a small vector algebra subroutine library, whose data types include vectors and matrices. I've implemented the usual cross product operator for: 3-vector cross 3-vector, ...
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Suppose the tensor equation $$T^{ij}S_{jk} = \delta^i_k$$ where $T$ and $S$ are different tensors of the same space. Suppose I know $S$, how would one go about solving said equation for $T$ in a ...
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I'm working on an exercise in linear algebra involving the multilinearity of the determinant. The task is: Let $𝐾$ be a field. Consider the determinant as a function: \begin{align} &\det : K^n \...
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It seems that the exterior derivative can be defined in the following way: The exterior derivative is a linear map $d: \Omega^k(M) \to \Omega^{k+1}(M)$ such that $\forall f_i \in C^{\infty}(M)$ \begin{...
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Let $V_1,\dots, V_t$ $(t\ge 2)$ be a $\mathbb{K}-$vector space having finite dimension. We define the tensor product of $V_1. \dots, V_t$ as $$V_1\otimes\cdots \otimes V_t:=\text{Mult}(V_1^*, \dots, ...
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In almost all places I look, the generalized Kronecker-delta symbol is defined as: $$\varepsilon_{i_1...i_p}\varepsilon^{j_1...j_p}=\delta _{j_1...j_p}^{i_1...i_p}$$ Which is the way I know it to be ...
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If V, W are vector spaces, its known what a n-linear function from $V^n \to W$ is. I personally see n-linear functions as a generalization of the notion of products $x_1*x_2*\dots *x_n$ in real ...
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Let $V_1, \ldots V_m$ be vectorspaces with $n_j := \dim(V_j)$ for all $j = 1, \ldots, m$ over some field $K$ and let $(e_j)_{j \in \{1,\ldots, n_j\}}$ be a basis of $V_j$ for all $j \in \{1,\ldots,m\}$...
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My homework defines lowering and raising operations (respectively) in the following way: $$\downarrow^a_b \colon T^r_s \to T^{r-1}_{s+1}, \quad (\downarrow^a_b A)(\omega_1, \dots, \omega_{r-1}, X_1, \...
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In the Clifford algebra, geometric algebra, and applications, the author introduced the following operations. Some of the relevant notation: $(Proposition)$ denote iverson bracket, i.e. $(P) = 1$ iff $...
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Let $\mathbf{F}$ be an arbitrary field. Given $k$ vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k \in \mathbf{F}^n$, write $\operatorname{Col}(\mathbf{v}_1, \ldots, \mathbf{v}_k)$ for the $n \times k$ ...
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I am working on a problem showing that a high-(uniform)-rank, low-dimension tensor of complex numbers can be decomposed into a matrix product state (MPS) and how this process can be parallelised ...
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I met a problem in tensor calculus: Suppose $\mathbf{A}$ and $\mathbf{B}$ are second-order tensors, with $\mathbf{A}$ being invertible, then we have the following tensor identity: $$\frac{\partial\...
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I am self-studying Fortney's book on differential forms (A visual introduction to differential forms and calculus on manifolds). I am solving question 3.11. The question is following: Let $\omega = a ...
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Let $ V $ be a vector space. Let $ g\in V^*\otimes V^* $ be a symmetric $ (0,2) $-tensor on $ V $, and call $ g $ also the bilinear mapping $ g\colon V\times V\to \mathbb R $ than corresponds ...
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The following is from Gowers's https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html: In fact, one can even do away with the condition that $X$ should be finitedimensional, as follows. If $f : V \times W \...
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on page 177 in book Orthogonal Polynomials of Several Variables 2nd.ed. is written , [where from prior definition(s) R is a vector in d dimensional space and member of Coxeter group W generated by ...
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Let's work on the real plane in polar coordinates $(\rho, \theta)$ and let $f(\rho,\theta)$ be a function of class $C^2$. My problem is the integration of the fowwing 2-form on the disc $B(0,r)=\{v\...
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spivak CoM exercise 4-30 could someone help me understand the hint? If I assume $w = \lambda d \theta + dg$ then $\int{c{R,1}} w = 2 \pi \lambda$. So the hint suggests I should integrate over $[0,1]$ ...
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Greub, Multilinear Algebra, section 4.1 defines an action of $S_k$ on $\bigotimes^k V$ by defining it on pure tensors as: $\sigma.(v_1\otimes ... v_k):= v_{\sigma^{-1}.1}\otimes ... v_{\sigma^{-1}.k}$ ...
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The answer to this question states that we have a nondegenerate pairing on a finite dimensional vector space $V$ $\wedge^k (V^*) × \wedge^k V \rightarrow \mathbb{K}$ Given on pure exterior vectors by: ...
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Suppose $A$ is an $n\times n \times n$ real symmetric tensor ($A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kji} \ \forall i,j,k$ in a basis). According to this paper, we can always write $A=\...
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I wished to prove that tensor product is associative in the sense that Given three finite-dimensional real vector spaces $X,Y,Z$, there exists a unique linear isomorphism $F:X\otimes Y\otimes Z\...
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Context Suppose $V$ is a $n$-dimensional vector space over $\mathbb{R}$. In our differential geometry class, we define the tensor product \begin{align} \overbrace{V\otimes \cdots \otimes V}^{k\text{ ...
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I am reading Differential Forms and Connections by Darling. Inner product of exterior powers is defined as follows: An inner product $\left\langle\cdot,\cdot\right\rangle$ is given for a vector space $...
Alex's user avatar
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The most general definition of tensor product of vector spaces I've found is through universal property. However, if I understand correctly, it defines two objects at the same time: the tensor product ...
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Here is a nice question. Consider a Euclidean vector space $V$ and denote by $e_v$ (resp. $i_v$) the endomorphism on $\Lambda^*V$ defined by $e_v\omega = v\wedge\omega$ (resp. $i_v= \iota_{v^*}$, ...
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I'm taking a course on multivariate real analysis this semester (the topics covered in the course are close to the topics in Analysis on Manifolds by James Munkres), The course is limited to the ...
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