Paper Info
Title
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 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. Our result generalizes the fact for scalars that for any complex logarithm and for all z is an element of C\{0}min(v is an element of(-pi,pi]) vertical bar Log(C)(e(-i upsilon)z)|(2) = vertical bar log vertical bar z vertical bar vertical bar(2), min(upsilon is an element of(-pi,pi]) vertical bar Re Log(C)(e(-iv)z)|(2) = vertical bar log vertical bar z vertical bar vertical bar(2).
Year
DOI
Venue
2014
10.1137/130909949
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Keywords
Field
DocType
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
Discrete mathematics,Orthogonal matrix,Frobenius matrix,Mathematical analysis,Unitary matrix,Matrix norm,Polar decomposition,Invertible matrix,Logarithm of a matrix,Hermitian matrix,Mathematics
Journal
Volume
Issue
ISSN
35
3
0895-4798
Citations 
PageRank 
References 
1
0.37
0
Authors
3
Name
Order
Citations
PageRank
Patrizio Neff122.79
Yuji Nakatsukasa29717.74
Andreas Fischle310.70