basic tensor (original) (raw)

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 thevector space 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. abasic tensor, if all but one of its values are 0, and the remaining non-zero value is equal to 1. For tuples A∈Ip andB∈Iq, we let

denote the characteristic array defined by

The type (p,q) characteristic arrays form a natural basis forTp,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 product on the left-hand side is performed by outer multiplication, and where A1⁢A2 on the right-hand side refers to the element of Ip1+p2 obtained by concatenating the tuplesA1 and A2, and similarly for B1⁢B2.

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

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 relations

add and multiply these symbols using coefficients in𝕂, and use

as a handy abbreviation for

We then interpret the resulting expressions as tensor arrays in theobvious 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 asuperscript, 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 vector

as

Similarly, a row-vector

can be written down as

In the case of a matrix

we would write