  We observe that the characteristic polynomial of a linearly perturbed
semidefinite matrix can be used to give the convergence rate of alternating
projections for the positive semidefinite cone and a line. As a consequence, we
show that such alternating projections converge at $O(k^{-1/2})$, independently
of the singularity degree. A sufficient condition for the linear convergence is
also obtained. Our method directly analyzes the defining equation for an
alternating projection sequence and does not use error bounds.
