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 $R in {mSO}(n)$ which minimize $$W: {m SO}(n) o mathbb{R}^+_0,quad W(R,;D) := ||{m sym}(RD -1)||^2$$ for a given diagonal matrix $D := {m diag}(d_1, ..., d_n) inmathbb{R}^{n imes n}$. The function $W$ subject to minimization is thereduced form of the Cosserat shear-stretch energy, which, in its general form,is a contribution in any geometrically nonlinear, isotropic and quadraticCosserat micropolar (extended) continuum model. We characterize the criticalpoints of the energy $W(R,;D)$, determine the global minimizers and the globalminimum. This proves the correctness of previously obtained formulae for theoptimal Cosserat rotations in dimensions two and three. The key to the proof isa characterization of the entire set of (possibly non-symmetric) real matrixsquare roots of (possibly non-positive definite) real symmetric matrices whichdoes not seem to be known in the literature.