On the factorization of matrices into products of positive definite ones

Abstract: The present work revisits and provides a new approach on a result by Charles Ballantine, on the factorization of a square matrix with positive determinant into a product of positive definite factors. {\em Ballantine-type} factorizations, that bound the number of positive definite factors, proved central in solving a basic, yet elusive control problem--the strong controllability of a linear system via control in the form of state feedback. Ballantine's result transcends control engineering, and highlights the little appreciated fact that rotations can be realized by the successive application of irrotational motions. Our approach is constructive and is based on the theory of optimal mass transport, specifically, it relates successive rotations of Gaussian distributions to corresponding optimal transport maps that constitute the sought factors.