Best approximation in Riemannian geodesic submanifolds of positive definite matrices

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold Sym(n, ℝ)⁺⁺ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold Sym(p, ℝ)⁺⁺ × Sym(q, ℝ)⁺⁺ block diagonally embedded in Sym(n, ℝ)⁺⁺ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when p ≤ 2 or q ≤ 2.