Details
Originalsprache | Englisch |
---|---|
Seiten (von - bis) | 28-42 |
Seitenumfang | 15 |
Fachzeitschrift | Linear Algebra and Its Applications |
Jahrgang | 449 |
Frühes Online-Datum | 26 Feb. 2014 |
Publikationsstatus | Veröffentlicht - 15 Mai 2014 |
Extern publiziert | Ja |
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.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
- Mathematik (insg.)
- Numerische Mathematik
- Mathematik (insg.)
- Geometrie und Topologie
- Mathematik (insg.)
- Diskrete Mathematik und Kombinatorik
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in: Linear Algebra and Its Applications, Jahrgang 449, 15.05.2014, S. 28-42.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › 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 -