Details
| Original language | English |
|---|---|
| Pages (from-to) | 28-42 |
| Number of pages | 15 |
| Journal | Linear Algebra and Its Applications |
| Volume | 449 |
| Early online date | 26 Feb 2014 |
| Publication status | Published - 15 May 2014 |
| Externally published | Yes |
Abstract
We show that the unitary factor Up in the polar decomposition of a nonsingular matrix Z=UpH is a minimizer for both∥-Log(Q *Z)∥-and∥-sym*(Log(Q *Z))∥- over the unitary matrices QεU(n) for any given invertible matrix ZεCn n×, for any unitarily invariant norm and any n. We prove that Up is the unique matrix with this property to minimize all these norms simultaneously. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.
Keywords
- Hermitian part, Majorization, Matrix exponential, Matrix logarithm, Minimization, Optimality, Polar decomposition, Unitarily invariant norm, Unitary polar factor
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
- Mathematics(all)
- Numerical Analysis
- Mathematics(all)
- Geometry and Topology
- Mathematics(all)
- Discrete Mathematics and Combinatorics
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In: Linear Algebra and Its Applications, Vol. 449, 15.05.2014, p. 28-42.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The minimization of matrix logarithms
T2 - On a fundamental property of the unitary polar factor
AU - Lankeit, Johannes
AU - Neff, Patrizio
AU - Nakatsukasa, Yuji
PY - 2014/5/15
Y1 - 2014/5/15
N2 - We show that the unitary factor Up in the polar decomposition of a nonsingular matrix Z=UpH is a minimizer for both∥-Log(Q *Z)∥-and∥-sym*(Log(Q *Z))∥- over the unitary matrices QεU(n) for any given invertible matrix ZεCn n×, for any unitarily invariant norm and any n. We prove that Up is the unique matrix with this property to minimize all these norms simultaneously. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.
AB - We show that the unitary factor Up in the polar decomposition of a nonsingular matrix Z=UpH is a minimizer for both∥-Log(Q *Z)∥-and∥-sym*(Log(Q *Z))∥- over the unitary matrices QεU(n) for any given invertible matrix ZεCn n×, for any unitarily invariant norm and any n. We prove that Up is the unique matrix with this property to minimize all these norms simultaneously. As important tools we use a generalized Bernstein trace inequality and the theory of majorization.
KW - Hermitian part
KW - Majorization
KW - Matrix exponential
KW - Matrix logarithm
KW - Minimization
KW - Optimality
KW - Polar decomposition
KW - Unitarily invariant norm
KW - Unitary polar factor
UR - http://www.scopus.com/inward/record.url?scp=84897716449&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1308.1122
DO - 10.48550/arXiv.1308.1122
M3 - Article
AN - SCOPUS:84897716449
VL - 449
SP - 28
EP - 42
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
SN - 0024-3795
ER -