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skew symmetry, anti-symmetry, alternance

One of the operations of tensor algebra, yielding a tensor that is skew-symmetric (over a group of indices) from a given tensor. Alternation is always effected over a few superscripts or over a few subscripts. A tensor with components is the result of alternation of a tensor with components , for example, over superscripts, over a group of indices if


The summation is conducted over all rearrangements (permutations) of , the number being or , depending on whether the respective rearrangement is even or odd. Alternation over a group of subscripts is defined in a similar manner.

Alternation over a group of indices is denoted by enclosing the indices between square brackets. Secondary indices inside the square brackets are separated by vertical strokes. For instance:

Successive alternation over groups of indices and , , coincides with alternation over the group of indices :

If is the dimension of the vector space on which the tensor is defined, alternation by a group of indices the number of which is larger than will always produce the zero tensor. Alternation over a given group of indices of a tensor which is symmetric with respect to this group (cf. Symmetrization (of tensors)) also yields the zero tensor. A tensor that remains unchanged under alternation over a given group of indices is called skew-symmetric or alternating over . Interchanging any pair of such indices changes the sign of the component of the tensor.

The operation of tensor alternation, together with the operation of symmetrization, is employed to decompose a tensor into simpler tensors.

The product of two tensors with subsequent alternation over all indices is called an alternated product (exterior product).

Alternation is also employed to produce sign-alternating sums of the form (*) with multi-indexed terms. For instance, a determinant with elements which commute under multiplication can be computed by the formulas


[1] P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian)
[2] D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian)
[3] J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)
[4] N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian)
How to Cite This Entry:
Alternation. L.P. Kuptsov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098