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Antisymmetric Part


Any square matrix A can be written as a sum

 A=A_S+A_A,
(1)

where

 A_S=1/2(A+A^(T))
(2)

is a symmetric matrix known as the symmetric part of A and

 A_A=1/2(A-A^(T))
(3)

is an antisymmetric matrix known as the antisymmetric part of A. Here, A^(T) is the transpose.

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

 A^(mn)=1/2(A^(mn)+A^(nm))+1/2(A^(mn)-A^(nm)).
(4)

The antisymmetric part of a tensor A^(ab) is sometimes denoted using the special notation

 A^([ab])=1/2(A^(ab)-A^(ba)).
(5)

For a general rank-n tensor,

 A^([a_1...a_n])=1/(n!)epsilon_(a_1...a_n)sum_(permutations)A^(a_1...a_n),
(6)

where epsilon_(a_1...a_n) is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example

 T^((ab)c)_([de])=1/4(T^(abc)_(de)+T^(bac)_(de)-T^(abc)_(ed)-T^(bac)_(ed)).
(7)

(Wald 1984, p. 26).


See also

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

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