Analytic Formulas for Alternating Projection Sequences for the Positive Semidefinite Cone and an Application to Convergence Analysis

We derive analytic formulas for the alternating projection method applied to 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 these 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 with singularity degree $2$.