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# Antisymmetric Part

Any square matrix can be written as a sum

 (1)

where

 (2)

is a symmetric matrix known as the symmetric part of and

 (3)

is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.

Any rank-2 tensor can be written as a sum of symmetric and antisymmetric parts as

 (4)

The antisymmetric part of a tensor is sometimes denoted using the special notation

 (5)

For a general rank- tensor,

 (6)

where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example

 (7)

(Wald 1984, p. 26).

Antisymmetric Matrix, Antisymmetric Tensor, Symmetric Matrix, Symmetric Part

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## References

Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984.

## Referenced on Wolfram|Alpha

Antisymmetric Part

## Cite this as:

Weisstein, Eric W. "Antisymmetric Part." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AntisymmetricPart.html