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Section 1.17 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)\label{equiv2}\tag{1.17.1} \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{1.17.2} \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{1.17.3} \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)\label{trID}\tag{1.17.4} \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\}\label{projective}\tag{1.17.5} \end{equation}
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 \(v\text{.}\)

Not surprisingly, all of this works over the other division algebras as well; (1.17.4) holds even over \(\OO\) since \(v\) has only 2 components, so that the computation takes place in a quaternionic subalgebra. Thus, (1.17.5) 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 (1.17.1), requires modification over the octonions, along the lines of the discussion in (((hopf))), 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{1.17.6} \end{equation}

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

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