Tensor contraction is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the trace is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a (1, 1) -tensor T i j can be contracted to a scalar through T i i , where the summation is again implied. When the (1, 1) -tensor is interpreted as a linear map, this operation is known as the trace.