Tensor taper

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The tensor taper or contraction is a mathematical term from linear algebra . It is a generalization of the trace of a linear mapping to tensors that are at least singly covariant and singly contravariant.


Let be a finite-dimensional vector space and be

the tensor space of -fold contravariant and -fold covariant tensors (short: -Tensors) over .

The linear mapping is called the tapering or contraction of a tensor (more precisely: contraction)

with and , which by

can be defined. There is an element of . Not every element of is of this form, but the elements of this form create the tensor space and the mapping is well defined. If one sets , then a tensor -th level becomes a tensor of level .


  • If one interprets a matrix as a simply co- and contravariant tensor, the taper of a matrix is ​​its trace. This can be seen very quickly if the matrix is ​​represented as a linear combination . Here one forms the basis of and that forms the dual basis of . If one now applies the function , one obtains. This shows that the tensor taper is a generalization of the trace operator known from linear algebra . For this reason, the mapping is also called track formation.

  • The Ricci tensor is obtained from the Riemann curvature tensor by tapering .


  • R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York NY et al. 1988, ISBN 0-387-96790-7 .

Individual evidence

  1. Ulrich E. Schröder: Special Theory of Relativity . German, Frankfurt am Main 2005, ISBN 3-8171-1724-8 , pp. 51 .