  We find analytic formulas for the alternating projection method for the cone
$\mathbb{S}^n_+$ of positive semidefinite matrices and an affine subspace. More
precisely, we find recursive relations on parameters representing a sequence
constructed by the alternating projection method. By applying the formulas, we
analyze the alternating projection method in detail and show that the upper
bound given by the singularity degree is actually tight when the alternating
projection method is applied to $\mathbb{S}^3_+$ and a $3$-plane whose
intersection is a singleton and has singularity degree $2$.
