Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

摘要

We consider the problem to determine the optimal rotations R ∈ SO(n) which minimize W : SO(n) → ?_0~+, W(R;D) = ||sym(RD - 1)||~2 for a given diagonal matrix D = diag(d_1,…, d_n) ∈ ?~(n×n) with positive entries d_i > 0. The objective function W is the reduced form of the Cosserat shear-stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy W(R;D), determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block-diagonal matrix with blocks of size at most two.