A LOGARITHMIC MINIMIZATION PROPERTY OF THE UNITARY POLAR FACTOR IN THE SPECTRAL AND FROBENIUS NORMS

作者：Neff Patrizio^*; Nakatsukasa Yuji; Fischle Andreas

来源：SIAM Journal on Matrix Analysis and Applications, 2014, 35(3): 1132-1154.

DOI：10.1137/130909949

• unitary polar factor
• matrix logarithm
• matrix exponential
• Hermitian part
• minimization
• unitarily invariant norm
• polar decomposition
• sum of squared logarithms inequality
• optimality
• matrix Lie-group
• geodesic distance

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 is an element of C-nxn 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 <= 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.

• 出版日期2014