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

Abstract

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 Log(Q * Z) 2 and its Hermitian part sym * (Log(Q * Z)) 2 over both R and C for any given invertible matrix Z ∈ C n×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 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. Our result generalizes the fact for scalars that for any complex logarithm and for all z ∈ C \ {0} min ϑ∈(−π,π] | Log C (e −iϑ z)| 2 = | log |z|| 2 , min ϑ∈(−π,π] | Re Log C (e −iϑ z)| 2 = | log |z|| 2 .

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