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

\[a_{ij}=-a_{ji}, \: i,j=1,2,3.\]