A Generalized Cauchy-Binet Formula and Applications to Total Positivity and Majorization

Abstract

This chapter discusses a generalized Cauchy–Binet formula and applications to total positivity and majorization. The identification and analysis of multivariate totally positive kernels, log concave densities, Schur-concave functions, and symmetric unimodal functions relies heavily on their conservation under convolution operators. An approach of wide scope incorporating many essential composition laws can be based on a generalized Cauchy–Binet formula. Thereafter, the Cauchy–Binet formula for matrix functionals plays an important role in the studies of determinants, permanents, and other classes of matrix functions.