A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

TL;DR

This paper proves that the unitary polar factor minimizes certain matrix logarithm norms, establishing its optimality in spectral and Frobenius norms and connecting it to geodesic distance in matrix space.

Contribution

It demonstrates the minimization property of the unitary polar factor for matrix logarithm norms in spectral and Frobenius norms, extending previous results.

Findings

The unitary polar factor minimizes the Log-based norms in spectral and Frobenius cases.

The minimization holds in any dimension for spectral norm and in 2-3 dimensions for Frobenius norm.

The result links the polar factor to geodesic distance in matrix space.

Abstract

The unitary polar factor $Q=U$ in the polar decomposition of the matrix $Z=UH$ is the minimizer for both $\| \mathrm{Log}(Q^* Z)\|^2$ and its Hermitian part $\| \mathrm{sym Log}(Q^* Z)\|^2$ over both $\mathbb{R}$ and $\mathbb{C}$, for any given invertible matrix $Z$ in $\mathbb{C}^{n\times n}$ and any matrix logarithm $\mathrm{Log}$, not necessarily the principal logarithm $\mathrm{log}$. We prove this for the spectral matrix norm in any dimension and for the Frobenius matrix norm in two and three dimensions. 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.