Contraction of a tensor. It occurs when a pair of literal indices (one a subscript, the other a superscript) of a mixed tensor are set equal to each other so that a summation over that index takes place (due to the Einstein summation convention). The result is another tensor whose rank is reduced by 2.
If a tensor is dyadic then its contraction is a scalar obtained by dotting each pair of base vectors in each dyad. E.g. Let be a dyadic tensor, then its contraction is ,
a scalar of rank 0.
E.g. Let be a dyadic tensor.
This tensor does not contract; if its base vectors are dotted the result is the contravariant metric tensor, , whose rank is 2.
