Antisymmetric Matrix

An antisymmetric matrix is a Matrix which satisfies the identity

$${\hbox{\sf A}}= -{\hbox{\sf A}}^{\rm T}$$ (1)

where ${{\hbox{\sf A}}}^{\rm T}$ is the Matrix Transpose. In component notation, this becomes

$$a_{ij} = -a_{ji}.$$ (2)

Letting $k = i = j$, the requirement becomes

$$a_{kk} = -a_{kk},$$ (3)

so an antisymmetric matrix must have zeros on its diagonal. The general $3\times 3$ antisymmetric matrix is of the form

$$\left[{\matrix{ 0 & a_{12} & a_{13}\cr -a_{12} & 0 & a_{23}\cr -a_{13} & -a_{23} & 0\cr}}\right].$$ (4)

Applying ${\hbox{\sf A}}^{-1}$ to both sides of the antisymmetry condition gives

$$-{{\hbox{\sf A}}}^{-1}{\hbox{\sf A}}^{\rm T} = {\hbox{\sf I}}.$$ (5)

Any Square Matrix can be expressed as the sum of symmetric and antisymmetric parts. Write

$${\hbox{\sf A}}= {\textstyle{1\over 2}}({\hbox{\sf A}}+{\hbox... ...{\textstyle{1\over 2}}({\hbox{\sf A}}-{\hbox{\sf A}}^{\rm T}).$$ (6)

But

$${\hbox{\sf A}}=\left[{\matrix{ a_{11} & a_{12} & \cdots & a... ...ots & \vdots\cr a_{n1} & a_{n2} & \cdots & a_{nn}\cr}}\right]$$ (7)

$${\hbox{\sf A}}^{\rm T} =\left[{\matrix{ a_{11} & a_{21} & \... ...ts & \vdots\cr a_{1n} & a_{2n} & \cdots & a_{nn}\cr}}\right],$$ (8)

so

$${\hbox{\sf A}}+{\hbox{\sf A}}^{\rm T}=\left[{\matrix{ 2a_{1... ... a_{1n}+a_{n1} & a_{2n}+a_{n2} & \cdots & 2a_{nn}\cr}}\right],$$ (9)

which is symmetric, and

$${\hbox{\sf A}}-{\hbox{\sf A}}^{\rm T}=\left[{\matrix{ 0 & a_... ... -(a_{1n}-a_{n1}) & -(a_{2n}-a_{n2}) & \cdots & 0\cr}}\right],$$ (10)

which is antisymmetric.

See also Skew Symmetric Matrix, Symmetric Matrix