2
votes
1answer
22 views

Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
0
votes
2answers
58 views

Multiplication and derivation of 3D matrix

I have $A(q)=\begin{bmatrix}q_1 &q_2 & q_3\\ 2q_1 &3q_2 & 4q_3\\ 2q_1 &3q_1 & 10\\ \end{bmatrix}\tag 1$ $ q= {\left(\begin{array}{c}q_1\\q_2\\q_3\\q_4\\q_5\\q_6 ...
3
votes
1answer
83 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
0
votes
1answer
32 views

A seemingly counterintuitive result on active and passive transformations of vectors

Let $\mathbf{v}$ be an element of a vector space with Euclidean $R^3$ as the underlying set. Assume the standard Cartesian basis $\{\mathbf{e^{(1)}, e^{(2)}, e^{(3)}}\}$ on it. Let $\mathbf{v^* = R ...
0
votes
0answers
13 views

Decompose matrix multiplication?

Sorry, I can break my imagination... I would like to decompose matrix multiplication into some tensor-like operations. As we know, matrix multiplication is done the following way $a_i^j b_j^k$ ...
0
votes
1answer
18 views

Rotation operator for a point in a coordinate system linearly derived from Cartesian coordinates

For some experimental and practical reason, I have created a new coordinate system in the form $$x^\prime_i=T_{ij}x_j$$ where $T_{ij}$ isn't a square matrix. $x_i$ is standard Cartesian coordinates, ...
5
votes
2answers
73 views

Exact meaning of “Not every matrix is a tensor”.

I've recently begun reading about tensors and am trying to understand the second order variety in the context of euclidean $\mathbb{R}^n$ with orthonormal basis {$e_1, e_2,\ldots, e_n$}. This seems ...
0
votes
1answer
48 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
1
vote
1answer
36 views

Matrix of a given operator $A \otimes A$

Let $V$ be a 3-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $e_{1}$, $e_{2}$, $e_{3}$ ...
1
vote
0answers
33 views

How is the multiplication between a multidimensional tensor with a matrix defined?

I am thinking this calculation in the following way but I am wondering if it is correct. Can anybody explain to me please? For example, I have a 3-way tensor $T^{u×i×t}$. How do I multiply this ...
0
votes
0answers
17 views

degrees of freedom (df) of a third order tensor

Does it make sense to simply unfold the tensor into a matrix and apply the df metric used for matrices? That is, a $\ n_1\times n_2 $ matrix of rank $\ r$ has df =$\ n_1r + (n_2 - r)r$ and so, a ...
3
votes
1answer
98 views

Transforming a matrix from cartesian to spherical coordinates

Consider a variable matrix $$\left[\begin{array}{ccc}a_{11}(x,y,z) \quad a_{12}(x,y,z) \quad a_{13}(x,y,z)\\ a_{21}(x,y,z) \quad a_{22}(x,y,z) \quad a_{23}(x,y,z)\\ ...
1
vote
0answers
26 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
0
votes
2answers
186 views

Multiplication of 3 matrices - Index vs. Matrix notation

i'm having a problem multiplicating 3 matrices in index notation. I know this should be trivial but i just can't figure it out. Is there any formula like $\ A'_{\mu\nu} = M_{\mu}^{\ ...
4
votes
2answers
88 views

How to generalize symmetry for higher-dimensional arrays?

@BrianM.Scott 's answer to this question Q: 3-dimensional array suggests that there is no standard concept of symmetry for 3-, 4-, N-dimensional arrays, in constrast to the case for 2-D arrays, as in ...
0
votes
0answers
45 views

Structure Tensor for the Algebra of 2x2 Matrices

Here's a question I need help understanding. Hope you can provide me some insight. Problem: Write the structure tensor for the algebra A of triangular $2\times 2$ matrices with real coefficients. ...
0
votes
2answers
78 views

Matrix/Tensor Operations

Suppose $A$ is an $m \times n$ matrix, and $B$ is an $n \times k$ matrix. Let $C$ be a tensor, where $$C(i,j,k) = A(i,j) + B(j,k)$$ What is a suitable (tensor) algebraic operation that summarizes ...
1
vote
1answer
158 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
0
votes
1answer
108 views

Linear System where Coefficient Matrix is a Kronecker Product

I have a system of linear equations where the coefficient matrix and right hand side is given by a Kronecker product: $(A_1 \otimes A_2) u = f_1 \otimes f_2$ My question: Is the solution simply ...
1
vote
1answer
95 views

Matrix tensor factorisation

Say we have a matrix $A$ expressed as the tensor $$A=\sum_{m=1}^Mx^{(m)}A^{(m)}$$ where $A$ and $A^{(i)}$ are $N\times P$ matrices and $x$ is a $M\times 1$ vector. I would like to decompose $A$ (or ...
15
votes
5answers
6k views

Differences between a matrix and a tensor

What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
1
vote
0answers
44 views

can I normalized the tensor rank in this way?

Is there such a thing as a "normalized tensor rank" for non-square tensors (i.e. a tensor with different sizes along each mode)? For example: If a 3rd order tensor (dimensions = 60 x 120 x 30) has ...
1
vote
1answer
78 views

Represent in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$

How can I represent this in a matrix form: $\sum_{ijl}(F_{il}-F_{jl})^2W_{ij}$ where all the entries are real and $W$ is a known(constant) matrix and $F$ is a rectangular matrix. When I say matrix ...
0
votes
1answer
33 views

Handling more than three indices/super indices, tensorial calculus

I need to code an equation such as the following one : $$ \frac{\partial u^j}{\partial q^i} = \frac{\partial \mathrm A^j_{pl}}{\partial q^i}\dot q^p \dot q^l + \frac{\partial \mathrm B^{jl}}{\partial ...
3
votes
1answer
591 views

“Inverse” of tensor product

I am trying to figure out something. I have a 4-tensor $\phi_{i \, j \, k \, \ell}$ and I know that $\phi = A \otimes B$, being $A$ and $B$ two matrices. With indices, I know this: $\phi_{i \, j \, k ...
2
votes
1answer
73 views

Tensor compact/matrix form.

I have got this tensor $S_{ij} = \frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$ Anyway I solve it for my problem and get $$ S_{ij} = \left( \begin{array}{ccc} 0 ...
1
vote
1answer
178 views

Gradient with respect to a matrix variable

I want to find the gradient of the function $\mathcal{F}_1$ with respect to the matrix $\mathbf{X}$ (differentiate with respect to $\mathbf{X}$): $$ \mathcal{F}_1 (\mathbf{X}; \mathbf{\lambda})= ...
1
vote
0answers
142 views

Multi-dimensional array decomposition

My question is about decomposing a muti-dimensional array into a product of matricies. To ask the question I will work towards the tensor, and then ask the question about the reverse process. Let ...
4
votes
1answer
192 views

What is the last index of a third-order tensor called?

In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
2
votes
1answer
505 views

Correct name for multi-dimensional array/matrix/tensor

What is the correct name for an n-dimensional array in mathematics? I have seen the following: nD-Matrix nD-Array nD-Tensor Which is the right way?
6
votes
2answers
658 views

Are matrices rank 2 tensors?

I know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular or ...
1
vote
1answer
279 views

Index notation clarification

Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw ...
4
votes
2answers
274 views

$\det(A \otimes B - B \otimes A) = 0$ why? Why $rk(M) = n^2-n$ ? Why x and -x in Spec(M) ?

Let $A$, $B$ be $n\times n$ matrices. It seems $\det(A \otimes B - B \otimes A) = 0$. Moreover it seems that the kernel of $A \otimes B - B \otimes A$ contains $n$ vectors. Here is MatLab code to ...
1
vote
2answers
474 views

is there a way to solve the following tensor equation?

I have the following tensor (takes a vector of length $m$ and returns a matrix $m \times m$): $C(y) = A \operatorname{diag}(A^T y ) A^{-1}$ for some invertible matrix $A$ of size $m \times m$ ($y$ ...
2
votes
1answer
44 views

Is there a particular name for a'long-small-small' tensor/array?

I'm thinking of a 3D array, with dimensions small,small,large. I've taken to saying 'sausage' as shorthand (and I'm sure there are worse NSFW descriptions) but is there a 'legitimate' description for ...
1
vote
1answer
194 views

Trouble deriving the Harris Corner Detection

I just started studying a small paper about the Harris Corner Detection. The problem is I don't understand how step 7 is derived from step 6. In step 7 the expression is expanded in a way that we get ...
17
votes
1answer
861 views

Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation

For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the ...