Table of Contents
What is symmetric and antisymmetric tensors?
Antisymmetric and symmetric tensors A tensor A that is antisymmetric on indices and has the property that the contraction with a tensor B that is symmetric on indices and. is identically 0.
Is the cross product a tensor?
A cross product is a vector, therefore it’s a tensor. To a physicist it’s particularly an object which transforms tensorially under changes of coordinates, ie, with one copy of the coordinate transformation matrix per index.
Are tensor products symmetric?
The category of vector spaces with tensor product is an example of a symmetric monoidal category. The universal-property definition of a tensor product is valid in more categories than just the category of vector spaces.
What is the difference between antisymmetric and symmetric relations?
A symmetric relation can work both ways between two different things, whereas an antisymmetric relation imposes an order. A symmetric relation can work both ways between two different things, whereas an antisymmetric relation imposes an order.
What does the tensor product represent?
The tensor product of both vector spaces V = VI ⊗ VII is the vector space V of the overall system. If the dimensions of VI and VII are given by dim(VI) = nI and dim(VII) = nII, the dimension of V is given by the product dim(V) = nInII. The tensor product is linear in both factors.
What does tensor product represent?
Tensor product of graded vector spaces[edit] Some vector spaces can be decomposed into direct sums of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).
What is the difference between tensor product and outer product?
In linear algebra, the outer product of two coordinate vectors is a matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra.
Is the totally antisymmetric tensor a pseudo-tensor?
Presumably, if the opposite were true then cross products etc. would be defined according to a left-hand rule, and would, therefore, take minus their conventional values. The totally antisymmetric tensor is the prototype pseudo-tensor, and is, of course, conventionally defined with respect to a right-handed spatial coordinate system.
What is the difference between parity inversion and totally antisymmetric tensor?
The totally antisymmetric tensor is the prototype pseudo-tensor, and is, of course, conventionally defined with respect to a right-handed spatial coordinate system. A parity inversion converts left into right, and vice versa, and, thereby, effectively swaps left- and right-handed conventions.
What is an antisymmetric contraction?
A tensor A that is antisymmetric on indices i and j has the property that the contraction with a tensor B that is symmetric on indices i and j is identically 0. (antisymmetric part). Similar definitions can be given for other pairs of indices.
Why is the cross product of two vectors not a vector?
It is therefore actually something different from a vector. We call it an axial vector. It turns out this this type of cross productof vectors can only be treated as a vector in three dimensions. In reality it is an antisymmetric tensor. Since there are only three independent numbers in this tensor, it can be cast as a vector.