The matched projection and geodesics of the Grassmann manifold

Given an idempotent operator $E$ in a complex Hilbert space ${\mathcal H}$, one can associate to it two orthogonal projections: - The polar decomposition $2E-1=(2P-1)|2E-1|$ provides an orthogonal projection $P$. That the unitary part in the decomposition of $2E-1$ is of this form, i.e., a selfadjoint unitary operator, is a remarkable observation done by G. Corach, H. Porta and L. Recht (see references below). - The question of which, among all orthogonal projections, is the one closest in norm to $E$, provides another projection, the so called {\it matched projection} $m(E)$, which answers this question. It was found by X. Tian, Q. Xu and C. Fu (see references below). In this paper we show that these projections coincide. Moreover, we show that there exists a unique minimal geodesic of the Grassmann manifold of ${\mathcal H}$ (the manifold of closed subspaces of ${\mathcal H}$) that joins $R(E)$ and $R(E^*)$. The orthogonal projection onto the midpoint of this geodesic, also coincides with $m(E)$.