Questions tagged [multilinear-algebra]

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

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Tensor Product $V \otimes W$ when $\dim V = 1$

Let $V$ be a vector space and $V'$ its dual, similarly for $W$ and $W'$. Given $v \in V$ and $w \in W$ we get an element of $V' \times W'$, written $v \otimes w$, as follows: for $\varphi \in V'$ and $...
Fly by Night's user avatar
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Determinant Formula for Wedge Product via Universal Property of Exterior Powers

I'm currently learning about differential forms in my analysis class, and I thought I'd dig a bit more into the linear algebra of exterior powers. I've seen the universal property of $\bigwedge^k(V)$: ...
Joshua Yagupsky's user avatar
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Does $V^* \otimes V^* \cong L(V, V; K)$ hold when $V$ is infinite dimensional?

Let $V$ be a $K$–vector space, let $V^*$ be its (algebraic) dual space, and let $L(V, V; K)$ denote the space of bilinear maps from $V \times V \to K$. Does $V^* \otimes V^* \cong L(V, V; K)$ hold ...
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If $A^TB$ is a symmetric matrix, then $XA=B$ has a symmetric matrix solution

Let $A,B\in\mathcal{M}_{m×n}(\mathbb{F})$ be rectangular matrices ($m\le n$) over an arbitrary field $\mathbb{F}$, such that $A$ is of full row rank. Moreover, $\require{enclose} \enclose{...
Aryan's user avatar
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Reference for the exterior product of operators

I'm looking for a textbook on linear/multilinear algebra where I can find an overview (definition & properties, preferably with proofs) of the exterior product of operators $$K_1\wedge...\wedge ...
Big Coconut's user avatar
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$\det(I-A)=\sum_k (-1)^k \operatorname{tr}(\wedge^k A)$

I have an equality which I am struggling to grasp: in an article the author says that $$ \det(I-A)=\sum_k (-1)^k \operatorname{tr}(\wedge^k A), $$ where $\wedge^k A$ is the map induced by $A$ on the $...
Nennee's user avatar
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Connection of Mixed Volume and Mixed Discriminant

I have thought about the connection of the mixed volume and the mixed discriminant for a while now but I got no satisfying answer out of the process. In detail: Mixed Volume: Let $\mathcal{K}^n$ be ...
Mathdealer's user avatar
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Solve constants in Coffin-Manson Equation [closed]

Solve for $b, c, x, y$, given the following: $$\begin{cases} x\cdot1000^b + y\cdot1000^c = .0068\\ x\cdot2000^b + y\cdot2000^c = .0058\\ x\cdot4000^b + y\cdot4000^c = .0050\\ x\cdot6000^b + y\...
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How can I decompose $x^2(x-y)+y^2(y-z)+z^2(z-x)$ into at least two factors?

Decompose the following expression into at least two factors, and if it does not decompose, you must prove with mathematical reasoning why it does not decompose (be sure to say with a mathematical ...
Arian Tajik's user avatar
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Unsure of construction of tensor product from bases

I've trouble understanding the definition of tensor products from the bases of the spaces which the operations is applied Given two vector spaces $V$ and $ W $ over the same field, with bases $ B_V $ ...
Tuxen's user avatar
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If I have two symmetric $k$-tensors $T$ and $S$ on a vector space $V$ such that $S(v,...,v) = T(v,...,v)$ for all $v$ how can I show that $S = T?$ [duplicate]

If I have two symmetric covariant $k$-tensors $T$ and $S$ on a vector space $V$ such that $S(v,...,v) = T(v,...,v)$ for all $v$ how can I show that $S = T?$ Apparently this question can be answered ...
Danlo's user avatar
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Intrinsic definition of tensor, tensor density, pseudotensors, even and odd tensors

Motivation I am trying to figure out what the intrinsic definition of different variants of "tensors" are supposed to be. I managed to find the transformation rules for the coefficients of ...
lightxbulb's user avatar
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Linear approximation using $2D$ Linear Interpolation

Probably it is a simple question. but even after several hours of research, I could not find anything relevant. I have a function $f(x,y) = xy$, with $x$ and $y$ that belong to bounded, continuous ...
Eventine's user avatar
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complex valued real-linear functionals

Let $V$ be a vector space over $\mathbb{R}$ and $V^{*}$ denote its dual space. Why is $V^* \otimes \mathbb{C}$ the space of complex-valued real-linear functionals on $V$?
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Are these R-homomorphisms necessary in this argument?

I just started to learn some multilinear algebra, that I lacked in my undergraduate program. I decided to do it with the book of Douglas Northcott. He defines what is going to be the solution for the ...
IJM98's user avatar
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What sort of transformation is a 3d cubic tensor?

I was curious about tensors and higher dimensional matrices. I know what a tensor is. But I can find next to no readable information about what or how to actually work with them? In short, what kind ...
Colonizor48's user avatar
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Levi-Civita symbol for cross product generalisation, geometric interpretation, orthogonal subspace

Cross Product in $\mathbb{R}^3$ In $\mathbb{R}^3$ one can compute the coordinates of the cross product between $b_2$ and $b_3$ using the Levi-Civita symbol as follows: $$b_{1,i_1} = \sum_{i_2i_3} \...
lightxbulb's user avatar
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Higher product rule for graded derivations

If $f^{i_1...i_p}$ and $g$ are smooth functions on some open set $U\subseteq\mathbb R^n$, with the former symmetrically indexed, then we have the higher product rule $$ \sum_{i_\bullet=1}^n\partial_{...
Bence Racskó's user avatar
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What is the relationship between: $\det(A + E[i,i]) = \det(A) + \det(A')$ and multilinearity?

Studying graph theory I came across the proof of Kirchof's theorem for maximal trees ("The number of generating trees of a graph $G$ is equal to the determinant of the reduced Laplacian Matrix of ...
MonkeyDL's user avatar
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Proving Basis of a type (1,1) Tensor

I am trying to teach myself tensor products, and I often see claims that the basis for a type (p,q) tensor is of the form: $e_{i_1}\otimes \ldots e_{i_q}\otimes e^{j_1}\otimes \ldots \otimes e^{j_p}$. ...
FreddyMcPolo's user avatar
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Solution of system of equations involving contravariant tensor is covariant tensor

I have an exercise in my linear algebra book as follows. Einstein notation is implied. A set of quantities $S_{ij}$ is defined in every coordinate system as the solution of the system of equations $T^...
Chordx's user avatar
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Kronecker Delta as a Tensor

Let $\delta^i_j$ be the Kronecker delta function, i.e. $1$ if $i=j$ and $0$ otherwise. Then, it is easy to verify that this value is a rank 2 mixed tensor of one covariant index and one contravariant ...
Chordx's user avatar
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Are there any nonzero continuous multilinear functions $\ell^\infty(\mathbb{R})\to\mathbb{R}$?

By multilinearity of $f:\ell^\infty(\mathbb{R})\to\mathbb{R}$ here I mean that, for every $(x_1,\cdots,x_n,\cdots)\in\ell^\infty$ and $i\in\mathbb{N}^*$, we have \begin{align*} f(x_1,\cdots,x_{i-1},...
Jianing Song's user avatar
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Orthogonal complements in exterior powers

Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge u_k, v_1\wedge \cdots \wedge v_k\rangle:=\det ([\langle u_i, v_j\...
Ian Morris's user avatar
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Tensor product in a noncommutative division ring

I have some problem understanding this proof by Qiaochu Yuan During the proof of the double centralizer theorem, he wrote the tensor product of two module in this way. Given $T\subset$ End$(A)$ and $T'...
Radagast's user avatar
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Is there are name for this class of multilinear functions related to the symmetric and alternating classes?

Two famous classes of multilinear functions are the symmetric and alternating multilinear functions, which satisfy for each $\sigma \in S_n$, $\sigma f = f$ and $\sigma f = \text{sgn}\sigma f$, ...
William's user avatar
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On the quadratic coalgebras

It is well known that a quadratic algebra $A(V,R)$ is the quotient of the free associative algebra $T(V)$ over a vector space $V$ by the two-sided ideal $(R)$ generated by $R\subseteq V^{\otimes 2}$, ...
Butters Stotch's user avatar
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Equivalent of unitary matrix with "spectrum" lying on the 2-sphere

The set of $n\times n $ unitary matrices has eigenvalues $\{\lambda_i\}_{i=1}^n$ which lie on the unit circle (or 1-sphere) $\mathbb{S}^1$. Is there an "object" A (with appropriate algebra) ...
Master Yogi's user avatar
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2 answers
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Proving that $A^{[ab]}B_{ab} = A^{ab}B_{[ab]}$

On this problem sheet, I am trying to show that for tensors (given a vector space V and it's dual $V^{*}$) $$A:V^{*} \times V^{*} \to \mathbb{R} \\ B: V \times V \to \mathbb{R}$$ that $A^{[ab]}B_{ab} ...
Taylor Rendon's user avatar
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Equivalence of p-vectors and skew-symmetric tensors

It is said (e.g. Lovelock and Rund) that p-vectors are equivalent to skew-symmetric tensors. However, a skew-symmetric form, such as $A_{ijk}$, is a specific form on $\mathbb{R}^n\times\mathbb{R}^n\...
Sam's user avatar
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Are there Plucker-like relations for the tensor product of two decomposable differential forms?

Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form $$ \mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge ...
WQE's user avatar
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1 answer
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trace of wedge product and cyclic property [closed]

Surprisingly, while their are similar but more advanced questions on this site, I don't see any answers to the basic version I am asking herein. If I am taking the trace of a wedge product of matrices,...
EEH's user avatar
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Trace of tensors in pseudo-Riemannian manifolds

I know that on a pseudo-Riemannian manifold $(M,g)$ a $(1,1)$ tensor field can be thought as an endomorphism of the tangent bundle. When one computes, for example, the trace of a $2$-covariant tensor, ...
Carlos Cabezas's user avatar
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Reference request: Book on theory of rank of matrices and multi-linear operators!

Is there a reference out there that only focus on (different)rank of matrices(with all kind of entries: real, complex, integers) and connects then further to ranks of tensors and further with the ...
Sarthak's user avatar
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3 votes
1 answer
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Dual of the bilinear space vs. the bilinear space of dual spaces

Let $X$, $Y$ be vector spaces (not necessarily finite dimensional) over field $k$. Is it always true $B(X,Y)^* \simeq B\left(X^*, Y^*\right)$? Here $B(X, Y)$ is the space of bilinear forms $b: X\times ...
user760's user avatar
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Zeroes of a $n$-variable boolean function of degree $d$.

Let $f$ be a non-constant boolean function of degree $d$ in variables $x_1,x_2,\dots,x_n$ where $d < n$. How do we prove that there exists a non-zero vector with at most $d+1$ many $1$ such that $f(...
Sagar Sawant's user avatar
3 votes
1 answer
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Notation for multilinear maps

This question is specifically in reference to Dimension of spaces of bi/linear maps but since it is a new question only indirectly related to that post (insofar as notation is concerned), I'll ask it ...
Michael Pugh's user avatar
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Polarization identity sesquilinear form

I don't understand why sometimes the polarization identity for sesquilinear form $m$ is written as $$ m(x,y)= \frac{1}{4}\sum_{n=0}^{3}i^{n}Q(x+i^{n}y) $$ with $Q(x)=m(x,x)$ and other times as $$ m(x,...
Davide Modesto's user avatar
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There is some generalitization of adjugate/adjoint matrix for tensors?

In some unrigorous physics texts the following "inverse" is used informally: $$\sum_{k,l=1}^n A_{ijkl} x_{kl} = b_{ij} \qquad \Rightarrow x_{kl} = \sum_{i,j=1}^n (A_{ijkl})^{-1} b_{ij}, \...
user910130's user avatar
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On Bishop's 'Tensor Analysis on Manifolds' Problem 5.10.6 (definition of Ricci tensor)

Here the complete problem statement (slightly modified): The Ricci tensor $R_{ij} \mathrm d x^i \otimes \mathrm d x^j$ of a connexion $D$ on a manifold is the tensor of type (0, 2) obtained by ...
Alfons Winkel's user avatar
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Applying Linear Regression to Compute Parameters of a Linear Combination of Two Lines

I'm dealing with a problem involving a linear combination on a bulk of lines. A standard line equation is: $$Ax+By+C=0$$ I define a "bulk" of lines as a linear combination of two boundary ...
Lu4's user avatar
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Symmetric Multilinear mapping is zero

Let $T:\underbrace{X\times X\times\dots\times X}_{n\ \text{times}}\to Y$ be a multilinear mapping that is symmetric. How can we prove that if: $T(x,x,...,x)=0, \forall x\in X$ then $T\equiv 0$? For $n=...
Bogdan's user avatar
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Classification of Multilinear Functions

In An Introduction to Manifolds, Tu defines: "Denote $V^{k} = V \times \dots \times V$ the Cartesian product of k copies of a real vector space V. A function $f: V^{k} \mapsto \mathbb{R}$ is k-...
Scott Sobolewski's user avatar
3 votes
1 answer
75 views

Understanding non-commutativity of tensor product when tensors are interpreted as R-valued linear maps

In the textbook Geometric Control of Mechanical Systems by Francesco Bullo and Andrew D. Lewis, I find the following: I am struggling to reconcile the definition of the tensor product given here with ...
Adam Sperry's user avatar
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Finding the change of basis matrix for a type (0,1) tensor

I am considering a tensor (in particular, the electric field), defined by $$E_m = g_{ij}^k c_{k\ell}^{ij}S_{\ell m} $$ Ultimately, this means that the tensor E is a rank 1, type (0,1) tensor, ...
Luk'yan Vilshansky's user avatar
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15 views

Bound symmetric tensor by its action on powers?

Let $A_{ijk}$ be a symmetric tensor (meaning that $A_{ijk}=A_{jik}=A_{kij}=\cdots$). Is there a bound of the form $$ |A_{ijk}| \leq C(d) \sup_{\|x\|_2=1} \left| \sum_{i,j,k=1}^d x_ix_jx_k A_{ijk}\...
felipeh's user avatar
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2 votes
1 answer
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A criteron for vanishing of Lie algebra cohomology

I am reading a criterion of Professor Serre for the vanishing of cohomology of Lie algebras in his paper "Sur les groupes de congruence des variétés abéliennes II". To prove this criterion, ...
Khainq's user avatar
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1 answer
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Sum of decomposable p-vectors is decomposable in exterior algebra

I'm studying Chapter 16 on Multilinear Algebra of MacLane and Birkhoff's Algebra and I'm a little confused about using decomposable vectors. The question I haven't been able to solve is this: Given $V$...
Braden Wilson's user avatar
4 votes
2 answers
355 views

Where is the magical sign change under change of basis ? Not pseudotensor?

I'm sorry for the long post but the this subject is confusing to me. Context: On one hand wiki talks about pseudovectors as if they are maps $\Phi:V^k \to V$ on the physical vector space with the ...
Physor's user avatar
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Free Vector Space of product space is isomorphic to tensor product of free spaces.

Let $S$ and $T$ be two arbitrary sets and consider the free vector space $C(S)$ and $C(T)$ generated respectively by $S$ and $T$. Show that $C(S \times T)$ is isomorphic to $C(S) \otimes C(T)$. I know ...
Mateo Soto Arango's user avatar

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