The tensor tag has no usage guidance.

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### Modifying tensor to be positive definite everywhere

Consider a (0,2)-tensor. It is known that it is positive definite somewhere and it is negative definite otherwise. Is there a theory how to "make" that tensor positive definite everywhere, while ...

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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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88 views

### 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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137 views

### 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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499 views

### Exponentiation of vector spaces?

This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...

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44 views

### 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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108 views

### 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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86 views

### Intuition for the tensor algebra? [closed]

As the question suggests, can someone give me their intuitions for working with the tensor algebra? Thanks in advance.
Here is my intuition/understanding for the tensor algebra. Given a ring $A$ ...

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### Restrictions of potential tensor fields to toric subgroups

Let $G$ be a compact connected nonabelian Lie group and let $f$ be a symmetric tensor field of order $m\geq1$ on $G$.
Let $T\subset G$ be a translate of a torus subgroup of $G$ with $\dim(T)\geq1$.
...

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393 views

### 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} = ...

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112 views

### 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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1k views

### 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)
...

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187 views

### Commutation of tensor products with inverse limits in a specific case

For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative ...

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102 views

### 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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416 views

### 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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506 views

### 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 ...

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871 views

### 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..."
...

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702 views

### Best known bounds on tensor rank of matrix multiplication of 3×3 matrices

Years ago I attended a conference where they taught us that matrix multiplication can be represented by a tensor. The rank of the tensor is important, because putting it into minimal rank form ...

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118 views

### 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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1k views

### Is there anyway to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there anyway (or any algorithm) ...

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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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310 views

### 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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269 views

### 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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206 views

### 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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### Representation theory of (anti)self-dual tensors

I am using usual physics notations and I guess the physics motivations of this question are obvious.
Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, ...

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529 views

### 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 ...

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### tensor/hypermatrix analogues of $GL(n,\mathbb{C})$?

Please excuse me if this question turns out to be incredibly silly for one reason or another.
Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? What I'm mainly ...

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296 views

### 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, ...

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### Who coined the name tensor and why?

Who coined the name "tensor" and why? What does the word "tensor" really mean, not the mathematical definition?

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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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### 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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### Tensor algebra question [closed]

1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$.
I can prove it using Hamel basis in $W$ but is ...

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### Maxwell Stress Tensor and Equations in Mathematician's Language [closed]

In my language, a differential two-form on $\mathbb{R}^4$ (viewed as a differentiable manifold with coordinates $t,x,y,z$) is a differentiable choice at each point of an alternating bilinear function ...

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### Why are matrices ubiquitous but hypermatrices rare?

I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...

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### Indexed tensor manipulation CAS

hello.
I am looking for tensor manipulation software that would allow me:
declare indices
declare results of contraction (or simplification rules)
allow algebraic simplifications and expansion
...

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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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### A generalization of Boolean matrix multiplication for order-3 tensors

The Boolean matrix product of two 0-1 $n \times n$ matrices $A$ and $B$ is the matrix $C$ defined as
$$C[i,j] = \vee_{k=1}^n (A[i,k] \wedge B[k,j]).$$ If $A = B$ and the matrix is an adjacency matrix ...