Lemma 5

Let E be a complex Hilbert space of dimension 2. Let us make the real vector space $\mathscr{L}(E)_h$ of Hermitian endomorphisms of E into a prehilbertian normed space by means of the norm $u\mapsto$ Tr($u^*u$)$^{1/2}$. The set of

orthogonal projectors of rank 1 of E is the sphere S of radius $1/$ 2 centred at $\frac{1}{2}1_E$ in the affine subspace of dimension 3 of endomorphisms of trace 1 in $\mathscr{L}(E)_h$.

Let F be the real affine subspace of $\mathscr{L}(E)_h$ formed by the elements of trace 1. The orthogonal projectors of rank 1 of E belong to F (Lemma 3, (ii)).

Let $u\in F$. We have $\|u-\frac{1}{2}1_E\|^2=$ Tr($u^2-u+\frac{1}{4}$) $=$ Tr($u^2$)$-\frac{1}{2}$. Consequently, $u\in S$ if and only if Tr($u^2$) $= 1$. Since 2 det($u$) $=$ Tr($u$)$^2-$ Tr($u^2$) $= 1-$ Tr($u^2$), this condition is equivalent to det($u$) $= 0$. By the Hamilton–Cayley theorem (A, III, p. 107, prop. 20), we therefore have $u\in S$ if and only if $u^2-u= 0$, which means that $u$ is a hermitian projector of rank 1 (loc. cit.), whence the result.