Proof That the Dot Product of a Vector With Itself Via a Skew Symmetric Matrix Is Zero
Theorem
Ifis skew symmetric then
Also if
then
is skew symmetric.
Proof
Sinceis skew symmetric
Ifthen
Ifthen
so
Conversely, suppose.
Let
Then
Similarly
Now take
Similarly
\[a_{ij}=-a_{ji}, \: i,j=1,2,3.\]