In this thesis, we study an approximation problem in the n x n real matrix space Mₙ (R). Two matrices A,B ∈ Mₙ(R) are said to be special orthogonal equivalent if there exist n x n special orthogonal matrices U and V such that A= UBV. A norm ║║ on Mₙ(R) is said to be special orthogonally invariant if ║UAV║=║A║for any n x n special orthogonal matrices U, V. Let B/~ be the equivalence class of B by special orthogonal equivalence and let conv B/~ denote its convex hull, i.e., the smallest convex set containing B/~. Given A, B ∈ Mₙ(R) and a special orthogonally invariant norm║║, we determine the quantities max{║A - X║: X ∈ conv B/~ } and min{║A - X║: X ∈ conv B/~ }. It turns out that we can find matrices B and Bₘ in conv B/~ such that ║A - Bₘ║ ≤ ║A - X║ ≤ ║A - B║ for all X ∈ conv B/~, and the pair of matrices work for all special orthogonally invariant norms. The problem is ultimately amount to solving an approximation problem in Rⁿ, which is of independent interests.