Bi‐invariant Integrals on GL( n ) with Applications

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.