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Antisymmetric Matrix

An antisymmetric matrix is a Matrix which satisfies the identity

\begin{displaymath}
{\hbox{\sf A}}= -{\hbox{\sf A}}^{\rm T}
\end{displaymath} (1)

where ${{\hbox{\sf A}}}^{\rm T}$ is the Matrix Transpose. In component notation, this becomes
\begin{displaymath}
a_{ij} = -a_{ji}.
\end{displaymath} (2)

Letting $k = i = j$, the requirement becomes
\begin{displaymath}
a_{kk} = -a_{kk},
\end{displaymath} (3)

so an antisymmetric matrix must have zeros on its diagonal. The general $3\times 3$ antisymmetric matrix is of the form
\begin{displaymath}
\left[{\matrix{
0 & a_{12} & a_{13}\cr
-a_{12} & 0 & a_{23}\cr
-a_{13} & -a_{23} & 0\cr}}\right].
\end{displaymath} (4)


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

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

Any Square Matrix can be expressed as the sum of symmetric and antisymmetric parts. Write
\begin{displaymath}
{\hbox{\sf A}}= {\textstyle{1\over 2}}({\hbox{\sf A}}+{\hbox...
...{\textstyle{1\over 2}}({\hbox{\sf A}}-{\hbox{\sf A}}^{\rm T}).
\end{displaymath} (6)

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


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

so
\begin{displaymath}
{\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],
\end{displaymath} (9)

which is symmetric, and


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

which is antisymmetric.

See also Skew Symmetric Matrix, Symmetric Matrix




© 1996-9 Eric W. Weisstein
1999-05-25