Questions tagged [multilinear-algebra]
For questions about the extension of linear algebra to multilinear transformations of vector spaces.
1,288
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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 $...
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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 $\bigwedge^k(V)$: ...
3
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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{...
- 1,459
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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
$$K_1\wedge...\wedge ...
- 412
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$\det(I-A)=\sum_k (-1)^k \operatorname{tr}(\wedge^k A)$
I have an equality which I am struggling to grasp: in an article the author says that
$$
\det(I-A)=\sum_k (-1)^k \operatorname{tr}(\wedge^k A),
$$
where $\wedge^k A$ is the map induced by $A$ on the $...
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Connection of Mixed Volume and Mixed Discriminant
I have thought about the connection of the mixed volume and the mixed discriminant for a while now but I got no satisfying answer out of the process. In detail:
Mixed Volume: Let $\mathcal{K}^n$ be ...
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Solve constants in Coffin-Manson Equation [closed]
Solve for $b, c, x, y$, given the following:
$$\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\...
- 21
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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 ...
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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 $ ...
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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 ...
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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 ...
- 2,027
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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 ...
- 27
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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 ...
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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 ...
- 349
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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:
$$b_{1,i_1} = \sum_{i_2i_3} \...
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Higher product rule for graded derivations
If $f^{i_1...i_p}$ and $g$ are smooth functions on some open set $U\subseteq\mathbb R^n$, with the former symmetrically indexed, then we have the higher product rule $$ \sum_{i_\bullet=1}^n\partial_{...
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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 ...
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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: $e_{i_1}\otimes \ldots e_{i_q}\otimes e^{j_1}\otimes \ldots \otimes e^{j_p}$. ...
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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^...
- 13
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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 ...
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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},...
- 1,333
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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 $$\langle u_1\wedge \cdots \wedge u_k, v_1\wedge \cdots \wedge v_k\rangle:=\det ([\langle u_i, v_j\...
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Tensor product in a noncommutative division ring
I have some problem understanding this proof by Qiaochu Yuan
During the proof of the double centralizer theorem, he wrote the tensor product of two module in this way.
Given $T\subset$ End$(A)$ and $T'...
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Is there are name for this class of multilinear functions related to the symmetric and alternating classes?
Two famous classes of multilinear functions are the symmetric and alternating multilinear functions, which satisfy for each $\sigma \in S_n$, $\sigma f = f$ and $\sigma f = \text{sgn}\sigma f$, ...
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On the quadratic coalgebras
It is well known that a quadratic algebra $A(V,R)$ is the quotient of the free associative algebra $T(V)$ over a vector space $V$ by the two-sided ideal $(R)$ generated by $R\subseteq V^{\otimes 2}$, ...
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Equivalent of unitary matrix with "spectrum" lying on the 2-sphere
The set of $n\times n $ unitary matrices has eigenvalues $\{\lambda_i\}_{i=1}^n$ which lie on the unit circle (or 1-sphere) $\mathbb{S}^1$.
Is there an "object" A (with appropriate algebra) ...
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Proving that $A^{[ab]}B_{ab} = A^{ab}B_{[ab]}$
On this problem sheet, I am trying to show that for tensors (given a vector space V and it's dual $V^{*}$) $$A:V^{*} \times V^{*} \to \mathbb{R} \\ B: V \times V \to \mathbb{R}$$
that $A^{[ab]}B_{ab} ...
- 2,216
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Equivalence of p-vectors and skew-symmetric tensors
It is said (e.g. Lovelock and Rund) that p-vectors are equivalent to skew-symmetric tensors. However, a skew-symmetric form, such as $A_{ijk}$, is a specific form on $\mathbb{R}^n\times\mathbb{R}^n\...
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Are there Plucker-like relations for the tensor product of two decomposable differential forms?
Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form
$$
\mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge ...
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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,...
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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, ...
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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 ...
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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 ...
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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(...
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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 ...
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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
$$
m(x,...
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There is some generalitization of adjugate/adjoint matrix for tensors?
In some unrigorous physics texts the following "inverse" is used informally:
$$\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}, \...
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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 ...
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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 ...
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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=...
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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-...
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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 ...
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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, ...
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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}\...
- 3,710
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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, ...
- 334
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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 $V$...
4
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2
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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 ...
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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 ...