basic tensor

The present entry employs the terminology and notation defined and described in the entry on tensor arrays. To keep things reasonably self-contained we mention that the symbol Tp,q refers to the vector spaceMathworldPlanetmath of type (p,q) tensor arrays, i.e. maps


where I is some finite list of index labels, and where 𝕂 is a field.

We say that a tensor array is a characteristic array, a.k.a. a basic tensor, if all but one of its values are 0, and the remaining non-zero value is equal to 1. For tuples AIp and BIq, we let


denote the characteristic array defined by

(εAB)j1jqi1ip={1 if (i1,,ip)=A and (j1,,jp)=B,0 otherwise.

The type (p,q) characteristic arrays form a natural basis for Tp,q.

Furthermore the outer multiplication of two characteristic arrays gives a characteristic array of larger valence. In other words, for


we have that


where the productPlanetmathPlanetmath on the left-hand side is performed by outer multiplication, and where A1A2 on the right-hand side refers to the element of Ip1+p2 obtained by concatenating the tuples A1 and A2, and similarly for B1B2.

In this way we see that the type (1,0) characteristic arrays ε(i),iI (the natural basis of 𝕂I), and the type (0,1) characteristic arrays ε(i),iI (the natural basis of (𝕂I)*) generate the tensor array algebra relative to the outer multiplication operationMathworldPlanetmath.

The just-mentioned fact gives us an alternate way of writing and thinking about tensor arrays. We introduce the basic symbols


subject to the commutation relationsMathworldPlanetmath


add and multiply these symbols using coefficients in 𝕂, and use


as a handy abbreviation for


We then interpret the resulting expressions as tensor arrays in the obvious fashion: the values of the tensor array are just the coefficients of the ε symbol matching the given index. However, note that in the ε symbols, the covariant data is written as a superscript, and the contravariant data as a subscript. This is done to facilitate the Einstein summation convention.

By way of illustration, suppose that I=(1,2). We can now write down a type (1,0) tensor, i.e. a column vectorMathworldPlanetmath




Similarly, a row-vector


can be written down as


In the case of a matrix


we would write

Title basic tensor
Canonical name BasicTensor
Date of creation 2013-03-22 12:40:37
Last modified on 2013-03-22 12:40:37
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 7
Author rmilson (146)
Entry type Derivation
Classification msc 15A69
Synonym characteristic array
Related topic TensorArray
Related topic Basis
Related topic Frame
Related topic SimpleTensor