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Section 8.3 Projective Lines

(This section was originally published in [1].)

There is another way to view \(vv^\dagger\text{,}\) namely as an element in projective space. Consider a pair of real numbers \((b,c)\text{,}\) and identify points on the same line through the origin. This can be though of as introducing an equivalence relation of the form

\begin{equation} (b,c) \sim (b\chi,c\chi)\tag{8.3.1} \end{equation}

where \(0\ne\chi\in\RR\text{.}\) The resulting space can be identified with the (unit) circle of all possible directions in \(\RR^2\text{,}\) with antipodal points identified. This is the real projective space \(\RP^1\text{,}\) also called the real projective line. But this space can also be identified with the squares of normalized column vectors \(v\text{,}\) that is,

\begin{equation} \RP^1 = \{vv^\dagger: v\in\RR^2, v^\dagger v=1\}\tag{8.3.2} \end{equation}

where we write \(\dagger\) instead of \(T\) for transpose, anticipating a generalization to the other division algebras.

The normalization condition can be written in terms of the trace of \(vv^\dagger\text{,}\) since

\begin{equation} \tr(vv^\dagger) = v^\dagger v\tag{8.3.3} \end{equation}

There is yet another way to write this condition, since

\begin{equation} (vv^\dagger)(vv^\dagger) = v(v^\dagger v)v^\dagger = \left(\tr(vv^\dagger)\right) \, (vv^\dagger)\tag{8.3.4} \end{equation}

Putting the pieces together, we obtain a matrix description of \(\RP^1\) in terms of \(2\times2\) real Hermitian matrices (\(\bH_2(\RR)\)), namely  1 

\begin{equation} \RP^1 = \{\XX\in\bH_2(\RR): \XX^2 = \XX, \tr\XX=1\}\tag{8.3.5} \end{equation}

Not surprisingly, all of this works over the other division algebras as well; (8.3.4) holds even over \(\OO\) since \(v\) has only 2 components, so that the computation takes place in a quaternionic subalgebra. Thus, (8.3.5) can be used to define the projective spaces \(\RP^1\text{,}\) \(\CP^1\text{,}\) \(\HP^1\text{,}\) and \(\OP^1\text{,}\) which are again known as projective lines.

However, the traditional definition, in terms of (8.3.1), requires modification over the octonions, as outlined in [1]. One possible choice would be

\begin{equation} \OP^1 = \{(b,c)\in\OO^2:(b,c)\sim\left( (bc^{-1})\chi,\chi \right),0\ne\chi\in\OO\}\tag{8.3.6} \end{equation}

with \(c=0\) handled as a special case.

For \(2\times2\) matrices \(\XX\text{,}\) the condition \(\tr\XX=1\) is needed to rule out the identity matrix, and ensures that \(\det\XX=0\text{.}\) This is enough to force one of the eigenvalues of \(\XX\) to be \(0\text{,}\) which in turn forces \(\XX=vv^\dagger\) for some \(v\text{.}\)