Explicit global minimization of the symmetrized euclidean distance by a characterization of real matrices with symmetric square

We determine the optimal orthogonal matrices R ∈ O(n) which minimize the symmetrized Euclidean distance W : O(n) → R, W (R ; D):= | | sym(RD - 1)| | ² , where 1 denotes the identity matrix and sym(X) = 1/2 (X + X^T ) is the symmetric part of X, for a given positive definite diagonal matrix D = diag(d₁, . . ., d_n) with distinct entries d₁ > d₂ > ⋯ > d_n > 0. The number of critical points depends on D and can grow faster than exponential in n. In the process, we prove and use a novel result of independent interest: every real matrix whose square is symmetric can be expressed as a block-diagonal matrix composed of blocks of size at most two by a suitable orthonormal change of basis.