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

2016/06/29 by Lev Borisov, Borisov, Lev, Andreas Fischle +3
Materials Science · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Characterization (materials science) #Combinatorics #Diagonal #Diagonal matrix #FOS: Physical sciences #Geometry #Isotropy #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Nonlinear Waves and Solitons #Nonlocal and gradient elasticity in micro/nano structures #Physics #Positive-definite matrix #Pure mathematics #Quadratic equation #Quantum mechanics #Square (algebra) #Square matrix #Symmetric matrix #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1606.09085

26 pages

openalex publication_date 2016/06/29 · arxiv created 2017/02/23 · arxiv updated 2017/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We consider the problem to determine the optimal rotations R \∈ rm\nSO(n) which minimize\n W:
rm SO(n)
to
mathbbR⁺₀,
quad W(R
,;D) := ||
rm sym(RD -\n1)||² for a given diagonal matrix D := rm diag(d₁, ..., d_n) \∈\n\ℝ^{n \× n}. The function W subject to minimization is the\nreduced form of the Cosserat shear-stretch energy, which, in its general form,\nis a contribution in any geometrically nonlinear, isotropic and quadratic\nCosserat micropolar (extended) continuum model. We characterize the critical\npoints of the energy W(R ,;D), determine the global minimizers and the global\nminimum. This proves the correctness of previously obtained formulae for the\noptimal Cosserat rotations in dimensions two and three. The key to the proof is\na characterization of the entire set of (possibly non-symmetric) real matrix\nsquare roots of (possibly non-positive definite) real symmetric matrices which\ndoes not seem to be known in the literature.\n

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