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Bi‐invariant Integrals on GL( n ) with Applications
Author(s) -
Schindler Werner
Publication year - 1995
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19951730117
Subject(s) - mathematics , invariant (physics) , lebesgue measure , invariant measure , borel measure , pure mathematics , probability measure , integrable system , measure (data warehouse) , diagonal , lebesgue integration , discrete mathematics , mathematical analysis , mathematical physics , database , computer science , geometry , ergodic theory
In this paper measures and functions on GL( n ) are called bi‐invariant if they are invariant under left and right multiplication of their arguments. If v is any bi‐invariant Borel measure on GL( n ), then there exists a unique Borel measure v * on D + ≥ ( n ), the set of all diagonal matrices of rank n with positive non‐increasing diagonal entries, such that\documentclass{article}\pagestyle{empty}\begin{document}$ \int\limits_{GL_{(n)} } {f(M) v(dM) = \int\limits_{D_{ + \ge (n)} } {f(D) v*(dD)} } $\end{document} holds for each v ‐integrable bi‐invariant function f :GL( n ) → IR. An explicit formula for v * will be derived in case v equals the Lebesgue measure on GL( n ) and the above integral formula will be applied to concrete integration problems. In particular, if v is a probability measure, then v * can be interpreted as the distribution of the singular value vector. This fact will be used to derive a stochastic version of a theorem from perturbation theory concerning the numerical computation of the polar decomposition.