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