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Symmetric 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.

The symmetric part of a tensor is denoted using parentheses as

 T_((a,b))=1/2(T_(ab)+T_(ba))
(4)
 T_((a_1,a_2,...,a_n))=1/(n!)sum_(permutations)T_(a_1a_2...a_n).
(5)

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)).
(6)

(Wald 1984, p. 26).


See also

Antisymmetric Matrix, Antisymmetric Part, Symmetric Matrix, Symmetric Tensor

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References

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

Referenced on Wolfram|Alpha

Symmetric Part

Cite this as:

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

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