Questions tagged [tensor]
The tensor tag has no usage guidance.
157 questions
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Is there a generalization of eigenvalues and eigenvectors to tensors?
Two perhaps ill-posed or just silly questions:
Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...
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Is there a relation between the Hessian matrix and the structure Tensor?
for a 2 dimensional image, i am interested to find a relation between the Hessian
$H = \begin{pmatrix} \frac{\partial^2 I}{\partial x^2}&\frac{\partial^2 I}{\partial x \partial y}\\ \frac{\partial^...
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Product of cotton-york tensor with ricci tensor
In the process of calculation of a problem in tensor analysis, I have encountered with an expression given by $$C^{ij}R_{ij}=0.$$
This is for a Riemannian manifold of dimension 3. Assuming the ...
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A kind of "Curvature tensor" for higher dimensional tensors
I begin my question with a multilinear question then I will consider two local smooth analogies:
Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...
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higher order analogues of sylvester's law of inertia?
Sylvester's law of inertia (here I quote wikipedia)
If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
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Strassen-like algorithm for Hadamard product of $2\times 2$ matrices
Let $A=\pmatrix{a_1& a_2\\a_3&a_4}, B=\pmatrix{b_1& b_2\\b_3&b_4}$ be two matrices. Let $C$ be the Hadamard product of $A$ and $B$.
$$C=\pmatrix{c_1& c_2\\c_3&c_4}=\pmatrix{...
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Looking for a tractable algorithm or formula for the determinant of a tensor
It is possible to define the determinant of a tensor.
We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map.
If $T:\{1,....,n\}^m\to \mathbb C$ ...
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Multi-dimensional permanent
Is there a particularly natural / "correct" way of generalizing permanents to tensors? (I mean of course, 'square' tensors.) There seem to be very few resources on the matter. There needs to be a ...
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Bounds on the tensor and border rank ratios of tensor unfoldings?
Consider $T_1 \in \mathbb{C}^{d_1 \times \cdots \times d_k}$ as an order-$k$ tensor of the format $(d_1, \cdots , d_k)$. Now, let $T_2$ be an unfolding of $T_1$ that is obtained by merging $\mathbb{C}^...
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Geometric Interpretation of Multiplication in Pure Cubic Number Fields and Beyond
I got interested in the question of possible geometric interpretations of the multiplication in algebraic number fields of degree $>2$ (with application to multiplication of units in the ring of ...
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Non Negative Tensor Tucker Decomposition Error Degradation
I have been working on iterative decomposition methods of tensors with non negativity constraints. I have noticed that $\textbf{N}$on negative $\textbf{T}$ensor $\textbf{F}$actorization "NTF" which is ...
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What is the largest tensor rank of $n \times n \times n$ tensor?
The tensor rank of a three dimensional array $M[i,j,k], i,j,k\in [1,\ldots,n]$ is the minimal number of vectors $x_i,y_i,z_i$, such that $M=\sum_{i=1}^d x_i\otimes y_i\otimes z_i$.
From dimension ...
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Finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry
The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/...
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Vanishing of determinant of Cotton York tensor
suppose $\Omega \subset \mathbb{R}^3$ is a Riemannian manifold with $g= dx_1^2 + dx_2^2 + c(x_1,x_2,x_3)dx_3^2$. Is it true that $det(CY)=0$?
Thanks
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Is a flattening rank a lower bound for the border rank?
Suppose $T \in V_1 \otimes \cdots \otimes V_k$ is a tensor, where each $V_i$ is a finite dimensional complex vector space. A $1$-flattening (or a flattening) is a realization of $T$ as a matrix in the ...
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Relation between curl and tensor divergence
The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas.
Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\...
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Derivative of eigenvalues w.r.t. a tensor
$E$ is a real, positive-definitive 3x3 symmetric tensor (I am thinking about the strain tensor in solid mechanics). We perform eigendecomposition and get:
$$E_p=\sum_{i=1}^{3}λ_iN_i⊗N_i$$
into its ...
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Doubt on the best low rank approximation of a symmetric tensor
I have a matrix $M\in\mathbb{R}^{n\times k}$, with $k<n$ whose columns $m_i$ are linearly independent.
So we have $M := [m_1|..|m_k]$.
From the columns of $M$ I can define the following matrix
$$
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Tell me something about these "component tensor" TQFT's
I noticed that there is a class of TQFT's that exists for every dimension $n\geq1$. It's probably well-known because it's quite simple, but I'm looking for a standard name or a better way to think ...
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Hessian as a tensor, multi-dimensional taylor series, and generalizations
The Hessian matrix $\{\partial_i \partial_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are two sets of ...
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Smoothness of a (given) global section of a vector bundle over G(2,6)
Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong $ det$(S^*)$, $S$ being the ...
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Proof about affine connections
I'm reading Nomizu & Sasaki's "Affine Differential Geometry: Geometry of Affine Immersions" and I'm having some trouble with Proposition 1.4.
I have an immersed surface in $M \hookrightarrow \...
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Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?
Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$).
Can we prove that the following problems are equivalent:
$$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
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Divergence of a second order tensor [closed]
I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct.
Suppose you would like to compute the components of a ...
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Recovering a smooth manifold from its tensor fields
1) Consider the algebra $\mathcal T (M) = \bigoplus \limits _{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with ...
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Mixed tensor index position significance
What is the significance of tensor index position?
For example the fourth order Riemann curvature tensor
\begin{align}
R^m_{ijk}
\end{align}
or
\begin{align}
R^{\phantom{i}m}_{i\phantom{m}jk}.
\end{...
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Obtaining the metric from the mixed Ricci tensor $R^i{}_j$
In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$).
But what do we know about ...
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Tensor of Relative Bases
Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...
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Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?
If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...
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Cannot multivectors be classified more easily than general tensors?
This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial ...
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Calculating the Riemann Christoffel tensor for a diagonal metric
I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$.
The Riemann-Christoffel tensor is given as
\begin{align}
R^m_{\phantom{m}ijk} = \...
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The Laplacian of an expression involving the Ricci tensor
While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...
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Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?
I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...
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Derivative of an eigenvector with respect to his own 3x3 real symmetric matrix
$\mathbf{C}$ is a real, positive-definitive 3x3 symmetric matrix (I am thinking about the right Cauchy-Green tensor in solid mechanics). We perform eigendecomposition and get:
$$\mathbf{C} = \sum_{i=...
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Petrov classification/Weyl scalars
There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup:
We want to show that Petrov type D (i.e. two principal null directions) ...
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Equations satisfied by the Riemann curvature tensor
It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
\end{...
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Global geometry measures for Riemannian manifolds
I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
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Mathematica package for supergravity and string theory
I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
CommunityBot
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Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?
Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
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Decomposition of order-3 tensors over the complex numbers
This is a question about decomposition of order-3 tensors. The survey Tensor Decompositions and Applications give a good account of recent developments in this area.
Let $T$ be an order-3 tensor, i....
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Levi-Civita symbol
Is the Levi-Civita symbol a tensor?
R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..."
$\epsilon_{jkq}=\epsilon^{jkq}=\...
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A property on the Green-St Venant strain tensor
Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient.
Show that
$u\in H^1(\Omega), E(u)\in L^r(\...
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Geometric interpretation of tracing [closed]
Let $(M,g)$ be a Riemannian manifold. Is it true that for any symmetric 2-tensor $\alpha$ we have:
$Trace_g(\alpha)=1/\omega_n\int_{S^{n-1}}\alpha(V,V)dvol(V)$
where $w_n$ is the volume of $S^{n-1}$?
...
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Number of linear independent equations
Is there any general rule to find the number of linearly independent equations such that
$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$
where $L_i$ is a ...
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How does a high order tensor irreducible decompose? [closed]
I know the two tensor case, but how about high order?
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Taylor's series for Lie groups
Let $G_1$ and $G_2$ be two (matrix) Lie groups, with $L(G_1)$ and $L(G_2)$ their respective Lie algebras.
I am interested to know if there is a well developed theory to approximate a (sufficiently) ...
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Null vector fields given Bondi metric
I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric
$g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$
with $d\Omega$-standard metric ...
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How to find the tensor product of modules that we don't know a basis for them?
Hi
I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[X] \otimes_{F} F[Y] \...
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How many flavors should a notational system offer for rank-1 tensors?
The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on ...
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Geometrical meaning of the Ricci Tensor and its Symmetry
Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume is that it has zero ...
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