  We consider the problem to determine the optimal rotations $R \in {\rm
SO}(n)$ which minimize
  $$W: {\rm SO}(n) \to \mathbb{R}^+_0,\quad W(R\,;D) := ||{\rm sym}(RD -
1)||^2$$ for a given diagonal matrix $D := {\rm diag}(d_1, ..., d_n) \in
\mathbb{R}^{n \times n}$. The function $W$ subject to minimization 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 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
a characterization of the entire set of (possibly non-symmetric) real matrix
square roots of (possibly non-positive definite) real symmetric matrices which
does not seem to be known in the literature.
