Any square matrix can be written as a sum
is a symmetric matrix known as the symmetric part of
is an antisymmetric matrix known as the antisymmetric part of .
is the transpose.
Any rank-2 tensor can be written as a sum of symmetric
and antisymmetric parts as
The antisymmetric part of a tensor is sometimes denoted using the special notation
For a general rank- tensor,
is the permutation symbol. Symbols for the
symmetric and antisymmetric parts of tensors can be combined, for example
(Wald 1984, p. 26).
See alsoAntisymmetric Matrix
, Antisymmetric Tensor
, Symmetric Part
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ReferencesWald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984.
on Wolfram|AlphaAntisymmetric Part
Cite this as:
Weisstein, Eric W. "Antisymmetric Part."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AntisymmetricPart.html