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.
The symmetric part of a tensor is denoted using parentheses as
 |
(4) |
 |
(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)).](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FSymmetricPart%2FNumberedEquation6.svg){kind=small} |
(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
Cite this as:
Weisstein, Eric W. "Symmetric Part." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SymmetricPart.html