| Popis: |
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 $n\leq 3$. The result shows that the unitary polar factor is the nearest orthogonal matrix to $Z$ not only in the normwise sense but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all $z\in\mathbb{C}{\setminus}\{0\} \min_{\vartheta\in(-\pi,\pi]}{|{\rm Log}_{\mathbb{C}}(e^{-i\vartheta} z)|}^2={|{\rm log}|... |
