Symmetric matrix

A square matrix MM is said to be symmetric if its transpose is equal to itself:

M=MTor equivalentlyMij=MjiM=M^{T}\quad\text{or equivalently}\quad M_{ij}=M_{ji}

It is said to be antisymmetric if its transpose is equal to the opposite of itself:

M=−MTor equivalentlyMij=−MjiM=-M^{T}\quad\text{or equivalently}\quad M_{ij}=-M_{ji}

Properties

Antisymmetric matrices (a.m.) are rather special. The sum of two a.ms is itself an a.m., and the product between a Scalar and an a.m. is again an a.m. This means that the sum and scalar product operations between a.ms are closed: this is sufficient condition to state that the space of all a.ms is a Vector space. More info on this can be found in Rotation > Rotation vector spaces.

Actually, it's something more than this. If we invoke the Commutator and take two a.ms Ω1\Omega_{1} and Ω2\Omega_{2}, we can calculate

([Ω1,Ω2])T=(Ω1Ω2−Ω2Ω1)T=Ω2T⏟−Ω2Ω1T⏟−Ω1−Ω1TΩ2T=Ω2Ω1−Ω1Ω2=−[Ω1,Ω2]([\Omega_{1},\Omega_{2}])^{T}=(\Omega_{1}\Omega_{2}-\Omega_{2}\Omega_{1})^{T}=\underbrace{ \Omega_{2}^{T} }_{ -\Omega_{2} }\underbrace{ \Omega_{1}^{T} }_{ -\Omega_{1} }-\Omega_{1}^{T}\Omega_{2}^{T}=\Omega_{2}\Omega_{1}-\Omega_{1}\Omega_{2}=-[\Omega_{1},\Omega_{2}]

From this we can claim that the commutator of two a.ms is itself an a.m, which means that the commutator is a closed operation in the a.m. vector space. This, combined with the vector space nature we found above, is sufficient to state that the space of antisymmetric matrices is a Lie algebra.