Tensor contraction

The Tensorverjüngung or contraction is a mathematical concept from linear algebra. It is a generalization of the trace of a linear map on tensors that are at least easy covariant and contravariant easy.

Definition

Be a finite dimensional vector space and let

The tensor of the times contravariant and covariant tensors times ( short - tensors ) on.

When tapering or contraction of a tensor (more precisely, contraction ) is called the linear map

With and which by

Can be defined. This course is an element of. Not every element of this form is, but the elements of the mold produce the tensor and the image is well-defined. So Substituting enough, it becomes a tensor -th stage, a tensor of rank.

Examples

• If one interprets a matrix as a simple co- and contra-variant tensor, the rejuvenation of a matrix is ​​its trace. This can see very quickly when the matrix as a linear combination represents. Here are the base of a and of the corresponding dual basis. Applying now the function that we obtain This indicates that the Tensorverjüngung is a generalization of the well known from linear algebra trace operator. For this reason, the picture is also called rutting.
• Obtained from the Riemannian curvature tensor by rejuvenating the Ricci tensor.