# Questions tagged [tensor-rank]

162 questions
Filter by
Sorted by
Tagged with
38 views

### Why can we substitute $V_{\mu \nu}$ to $V_{\mu ; \nu}$ while inducing contracted Bianchi identity?

After $( A_\mu B_\nu )_{; \sigma ; \rho} - ( A_\mu B_\nu )_{; \rho ; \sigma} = A_\alpha B_\nu R^\alpha_{\mu \rho \sigma} + A_\mu B_\alpha R^\alpha_{\nu \rho \sigma}$ where $A_\mu B_\nu$ is outer ...
83 views

### Did I perform the partial derivative correctly?

I have the following expression: $\gamma^s=\sum_{s'}(X^{ss'})^{-1}m^{s'}(C-\sigma\otimes I)\cdot\varepsilon$ Where $s$ is the number of slip systems in the material and can be any integer. For this ...
27 views

44 views

### Algebra equation for 3 rank tensor

Suppose I work in $4$ dimensions. I have an algebraic equation in the following form, which contains a 3 rank tensor $X ^{\alpha \lambda \mu }$ X ^{\alpha \lambda \mu }\eta ^{\beta \...
• 19
70 views

### Understanding the Definition of a Rank-1 Tensor

A tensor is nothing but a multidimensional array. We can think of an $n-mode$ tensor as a structure whose each element has to be referred with the help of $n$ indices or $n$ axes. Now, while reading ...
• 2,704
37 views

### 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 ...
• 29
1 vote
122 views

### Prove a tensor identity

Prove the tensor identity: $$e_{ij}^2-\frac{1}{3}e_{kk}^{2}=(e_{ij}-\frac{1}{3}\delta_{ij}e_{kk})^{2}$$ This equality we used it to prove the conservation of energy equation although i don't see ...
• 153
137 views

### Higher order Tensor computations

I have a $D$-way tensor H of dimension $I \times I \times \dots \times I$ ($D$ times), that represent the coefficients of a polynomial. For better understanding, I provided an image of 3-way tensor ...
• 57
32 views

### How to represent a bivector/bi-covector or more complex tensors (of rank 1,2/2,1/2,2 or even higher) using numbers?

If I understand correctly, the rank 0,0 tensor is just a scalar, rank 0,1 tensor is a vector represented as a column, rank 1,0 is a covector represented as a row, rank 1,1 is a linear transformation ...
14 views

### Symmetric Trilinear map for proving Newton's method on self-concordance functions

The following claim is used for proving the quadratic convergence phase of Newton's method on self-concordance functions. I have a very long and non-intuitive proof using Lagrange Multipliers. I ...
• 320
19 views

### show that the mixed derivative of zero order tensor gives second order tensor [duplicate]

I am trying to prove that for a zero order tensor, $f$ the expression $\partial^2f \over \partial x_i \partial x_j$ is a second order tensor, yet struggle to do so. I'd appreciate your help.
124 views

### Tensorial Representation of a Complex Network (Questions on Tensors)

INTRODUCTION TO QUESTION 1 Some authors proposed a tensorial representation of complex networks (for both single layer networks and multilayer networks). One reference paper for this topic is this one:...
• 329
35 views

### How do you determine the rank of the differential of an immersion?

I'm trying to prove that some mapping of a manifold is also a manifold using the constant rank theorem but I don't really understand the meaning of rank in this context. As I understand it, the ...
1 vote
99 views

### Does "$\Gamma_{k i j}=\Gamma^m{ }_{i j} g_{m k}$" mean tensor-contraction or multiplication?, i.e. $\Gamma_{k i j}=\Gamma^m{ }_{i j} \cdot g_{m k}$?

Does $\Gamma_{k i j}=\Gamma^m{ }_{i j} g_{m k}$ mean tensor-contraction or does it mean multiplication, i.e. $\Gamma_{k i j}=\Gamma^m{ }_{i j} \cdot g_{m k}$ An alternative definition, I cannot ...
45 views

### Can a rank 1 tensor represent the same information that a rank n tensor does?

So, I was studying quantum mechanics, and a question came up. In quantum mechanics, we work in a Hilbert space, and the basis that we choose can represent all the information that we can have in order ...
386 views

### What is the reasoning/need for writing tensor indices like $T^i{\,}_j^{{\,\,}k}$ instead of $T_{j}^{ik}$ (or, $A^i_{\,\,\,j}$ instead of $A^i_j$)?

In one of the comments to this previous question I wanted to know why one of the users that gave an answer kept writing connection terms as ${\Gamma}^{\,\,\,l}_{h\,\,k}$. Until I read that answer I ...
• 237
79 views

• 734
1 vote
268 views

### Rank 1 tensors, how to describe them? (specific case)

I want to undestand a specific case. I consider two $\mathbb{C}$-vectorial spaces, $\mathbb{C}^2$ both. Then, I want to work with $\mathbb{C}^2\otimes \mathbb{C}^2$. Now, I consider basis for each ...
• 219
196 views

### Dimension of the variety of rank 1 decompositions of a matrix

Let $A\in\mathrm{GL}_n(\mathbb{C})$, and let's consider its decompositions into a sum of rank 1 matrices $$A=\sum_{i=1}^t A_i,\ \text{rank}(A_i)=1,$$ where $t\geqslant n$. When $t=n$, Wikipedia claims ...
• 4,018
63 views

### Rank of Matrix Multiplication tensor $\langle1,n,1\rangle$

$\newcommand{\rank}{\operatorname{rank}}$I am reading through Belzer's Fast Matrix Multiplication, available here. I want to prove tjat $$\rank(t)=\rank(\langle1,n,1\rangle)=n$$ In "usual tensor ...
I am reading through Fast Matrix Multiplication by Markus Blaser. I am trying to prove Lemma 5.3 from page 19. It states the following: For any tensor $T\in \mathbb{F}^{n\times m\times t}$, and any ...