Why must norms of basis vectors square to a value in the set {-1, 0, 1}?

Projective GA
1

Hello,

I am new to geometric algebra and saw a comment on another post stating that the square of an orthogonal basis vector belongs to the set {-1, 0, +1}.

Is there a mathematical reason why ei2 have to map to an element in the set {-1, 0, +1} or is just a convention that proves to have utility?

2

Basis vectors don’t need to be unit length or orthogonal to each-other. But if they aren’t, you can then pick a new basis for the same space which does satisfy those constraints, and is usually more convenient to work with.

1 Like
3

Hi @kend,

you can get a look at the algebra \mathbb{R}^{4,4} of Goldman and Mann. The basis vectors still square to 0 but the metric is full of 0.5 :

\begin{array}{cccccccc} 0 & 0 & 0 & 0 & 0.5& 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.5 &0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0.5 &0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5\\ 0.5 & 0 & 0& 0 & 0 & 0 & 0 & 0\\ 0 & 0.5 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0.5& 0 & 0& 0 & 0 & 0\\ 0 & 0 & 0 & 0.5 & 0 & 0 & 0 & 0\\ \end{array}

They could have set only “1” instead of “0.5”, this choice is motivated to avoid power of 2 in the versors.

1 Like