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 arg uments. If v is any bi-invariant Borel measure on GL(n), then there ex ists a unique Borel measure v on D-+greater than or equal to(n), the set of all diagonal matrices of rank n with positive non-increasing di agonal entries, such that [GRAPHICS] holds for each v-integrable bi-in variant function f:GL(n) --> R. An explicit formula for v will be der ived in case v equals the Lebesgue measure on GL(n) and the above inte gral formula will be applied to concrete integration problems. In part icular, if v is a probability measure, then v can be interpreted as t he distribution of the singular value vector. This fact will be used t o derive a stochastic version of a theorem from perturbation theory co ncerning the numerical computation of the polar decomposition.