A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

@article{Neff2014ALM,
  title={A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms},
  author={Patrizio Neff and Yuji Nakatsukasa and Andreas Fischle},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2014},
  volume={35},
  pages={1132-1154}
}
The unitary polar factor $Q=U_p$ in the polar decomposition of $Z=U_p \, H$ is the minimizer over unitary matrices $Q$ for both $\|{\rm Log}(Q^* Z)\|^2$ and its Hermitian part $\|{{\rm sym}{_{_*}}\!}({\rm Log}(Q^* Z))\|^2$ over both $\mathbb{R}$ and $\mathbb{C}$ for any given invertible matrix $Z\in\mathbb{C}^{n\times n}$ and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any $n$ and for the Frobenius matrix norm for $n\leq… 

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