0
votes
0answers
13 views

Index notation for inverse matrices

I have a question: There is an standard way to write the inverse of a matrix in index notation?. The reason is that I don't want to write $(A^{-1})_{ij}$ or $(A^{-1})_i^j$ or $(A^{-1})^{ij}$ using ...
3
votes
1answer
29 views

How to interpret tensor form PDE in terms of matrix algebra

From this mathwork page "c for system", the usual second order PDE is written in tensor form: $$ -\nabla\cdot(\mathbf{c} \otimes \nabla \mathbf{u})+\mathbf{a}\mathbf{u}=\mathbf{f} $$ and ...
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
1answer
47 views

Decomposition of Tensor into a Product of Tensors

I was working on a text that made use of the assumption that, for some rotation matrix $a_{ij}$, and some tensor $U$, the tensor under rotation is represented by $U'_{\alpha}=a_{\alpha i}U_i$. It made ...
3
votes
1answer
87 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
56 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
0
votes
1answer
31 views

Is it necessary for a linear map to be an automorphism to allow polar decomposition?

Bowen and Wang's Introduction to Vectors and Tensors I (pg. 168) states a general form of the polar decomposition theorem as Every automorphism A has two unique multiplicative decompositions $$ ...
0
votes
3answers
29 views

Tensor tranformation between basis?

If I am the basis vector $e_i$ into another basis to get $e'_j$ I use: $$e'_j=S_{ij}e_i$$ My text book says that $S_{ij}$ is the ith component of the vector $e'_j$ with respect to the unprimed basis. ...
2
votes
1answer
66 views

Which Subspaces do Antisymmetric Tensors Represent?

So antisymmetric tensors represent volumetric subspaces (I've asked this here instead of on phys.stackexchange because it seems like more of a math question)? How exactly would one know WHICH ...
7
votes
0answers
79 views

Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
0
votes
1answer
16 views

sum representation by/ determinant of elementary tensors

Consider the bijective linear map: $\alpha : K^2 \otimes K^2 \to Mat(2 \times 2,K), \alpha(v \otimes w) = vw^t, v, w \in K^2$ , where $K$ is an arbritrary field. First I want to show that every ...
1
vote
1answer
38 views

matrix inverse in tensor notation

Suppose there is a matrix $A$ that transforms vectors, $$ Y = A x $$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ ...
0
votes
1answer
23 views

Linear transformation from endormorphism to real number

For a finite dimensional vector space $V$, is there a linear transformation between its endomorphism and real number, please? I suspect that since the element of the endomorphism can be represented by ...
0
votes
1answer
64 views

Confusing question on tensors

A tensor of rank $4$ satisfies $T_{ijkl}=T_{jilk}=-T_{jikl}$ and $T_{ijij}=0$. I need to show that: $$T_{ijkl}=-\varepsilon_{ijp}\varepsilon_{klq}T_{rqrp}$$ Could someone offer a hint? I have tried ...
5
votes
1answer
68 views

The derivative of the determinant of a Kronecker product

For an invertible matrix $A$, we have the identity \begin{align} \dfrac{\partial \det A}{\partial A} = \det A (A^{-1})^T \end{align} where the $T$ denotes the transpose operation. How does this ...
2
votes
0answers
23 views

Index notation interpretation

I'm having some confusion with index notation and how it works with contravariance/covariance. $(v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j$ $(v_{new})^i=J^i_{\ ...
0
votes
2answers
45 views

Tensor Product over a ring

Given Two Fields $F,K$, and two vector spaces $V,W$ over $F$, what does tensor product $$V\otimes_{K} W$$ mean? I am not certain whether this is defined in general. I came across it in cases wheh ...
1
vote
1answer
20 views

When is a symmetric 2-tensor field globally diagonalizable?

Suppose that $\mathbb{R}^n$ has a Riemannian metric $g$. Let $h$ be a smooth symmetric 2-tensor field on $\mathbb{R}^n$. At any point $p \in \mathbb{R}^n$, there is a basis of $T_p \mathbb{R}^n$ in ...
2
votes
2answers
48 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
2
votes
1answer
103 views

using the kronecker product and vec operators to write the following least squares problem in standard matrix form

I have a least squares problem with the following form: $$ \min_\mathbf{X} ~ \left\| \sum_{i=1}^n \mathbf{u}_i^\top \mathbf{X} \mathbf{v}_i - b_i \right\|^2 $$ where $\{\mathbf{u}_i\}_{i=1}^n$ and ...
0
votes
2answers
52 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
0
votes
0answers
19 views

Alternative operator is a homomorphism?

Let $V$ be a real vector space of dimension $n$, for a (real valued) tensor $f$ of order $r$, define the alternative operator $A$ by $$(Af)(v_1,\cdots, v_r)=\frac{1}{r!}\sum_{\sigma\in ...
1
vote
1answer
80 views

Tensor Einstein summation notation

I have two tensors $A^i$ and $B_j$ with components $(2,3,4)$ and $(1,2,3)$ respectively. What is the difference between $A^i B_i$ and $A^i B_j$? Is it just: $A^i B_i = 2+6+12 = 20$ $A^i B_j =$ $ ...
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
188 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}^{\ ...
2
votes
2answers
88 views

Square vs non-square tensors?

In mathematics, tensors are objects that operates on vector space. In physics or engineering, tensors usually operates on one vector space and its dual space: $V^{*} \times V^{*} \times V^{*} \times ...
3
votes
0answers
66 views

dot product between vector and matrix

In my book on fluid mechanics there is an expression $$ \boldsymbol{\nabla}\cdot \boldsymbol{\tau}_{ij} $$ where $\boldsymbol{\tau}_{ij}$ is a rank-2 tensor (=matrix). Given that ...
6
votes
1answer
329 views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
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 ...
1
vote
1answer
156 views

Trace of tensor product

I am currently struggeling with the following problem: Imagine that you have 4 vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ and then you want to calculate $\hbox{trace}(v_4 \otimes v_3 \cdot ...
1
vote
0answers
44 views

Derivative of a tensor

I have a rank-2 tensor given by $$ P_{\alpha \beta} = p\delta_{\alpha \beta} + (u_1^2, u_1u_2 ; u_1u_2, u_2^2) $$ whose derivative with respect to $x_{\alpha}$ I would like to find. According to my ...
1
vote
2answers
117 views

Annihilator of a Tensor

This is a question I have trouble understanding, hope you can clarify this to me. Problem: Find the annihilator of the tensor $e_1\wedge e_2+e_3\wedge e_4$ 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 ...
2
votes
0answers
57 views

Factoring tensors by linear transformation

Let $T_n$ be tensors $\left( V^n \rightarrow \mathbb{R}\right)$ of order $n$. Now let $\sim$ be equivalence on $T_n$ that $A\sim B$ iff there is regular matrix $M$ that $A\circ M = B$. By $A\circ M$ ...
2
votes
1answer
137 views

How to translate between differential forms and tensor index notation

The books on Manifold theory & geometry that I studied introduce connection and curvature in the language of differential forms. But Physics books on the other hand like (General Relativity by ...
0
votes
0answers
29 views

When can I recover a (full) tensor-contraction after normalising partial contractions.

I have rank-${1 \brack n}$ tensor $R_{abc....}$ and a rank-${n \brack 1}$ tensor $S^{ABC...}$. Obviously their contraction $$u = R_{abc....}S^{abc...}$$ is a scalar (complex in my case). I can ...
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
0answers
101 views

Rank-2n tensor algebra eigenvalue equation

Im interested in resources and work done on the eigenvalue equation for rank-2n tensors: $$ M_{ij}A_{j} = \lambda A_{i} \\ $$ $$ M_{ijkl}A_{kl} = \lambda A_{ij} \\ $$ $$ M_{ijklmn}A_{lmn} = \lambda ...
1
vote
0answers
382 views

What are these physicists talking about? Dyadic green function?

I am interested in a mathematical explanation(in the sense that you say for example: is it a mapping from A to B) what a dyadic green function and the unit dyad actually is? I am reading this as ...
10
votes
5answers
483 views

Book on tensors

Can anyone recommend me a book on tensors with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I want to understand why tensors were ...
0
votes
1answer
56 views

Suficient condition for tensor product of vector spaces..

Can anyone help me showing the following: Let $E$, $F$, $G$ and $H$ vector spaces and $\varphi:E\times F\rightarrow G$ a bilinear map. If for every $\psi:E\times F\rightarrow H$ bilinear there is an ...
6
votes
1answer
767 views

Vorticity equation in index notation (curl of Navier-Stokes equation)

I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation: $$ {\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = ...
4
votes
1answer
265 views

Tensors as mutlilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely $$ V\otimes W := L_2(V^*\times W^*,\Bbb F) $$ I am also aware that this space is isomorphic to the tensor ...
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 ...
1
vote
1answer
130 views

Is the third order term of a Taylor approximation a covariant tensor of rank 3 restricted to (h,h,h)?

Recently reading Peter Lax's linear algebra text, he had a very concise way of writing the Taylor approximation up to second order: $f(x+h) = f(x) + l(h) + \frac{1}{2}q(h) + \|h\|^2 \epsilon(\|h\|)$, ...
2
votes
3answers
56 views

notation question (bilinear form)

So I have to proof the following: for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form. ...
1
vote
1answer
40 views

Finding an “inverse” of a deviatoric tangent

I have have a material model, defining the deviatoric stress for a nonlinear fluid: $\boldsymbol{\sigma}_{\mathrm{dev}} = f(\dot{\boldsymbol{\varepsilon}}_{\mathrm{dev}})$ Now I wish to find the ...
0
votes
0answers
69 views

Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
1
vote
0answers
143 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 ...