The factorization of matrices into products of positive definite factors

摘要

Positive-definite matrices materialize as state transition matrices of linear time-invariant gradient flows, and the composition of such materializes as the state transition after successive steps where the driving potential is suitably adjusted. Thus, factoring an arbitrary matrix (with positive determinant) into a product of positive-definite ones provides the needed schedule for a time-varying potential to have a desired effect. The present work provides a detailed analysis of this factorization problem by lifting it into a sequence of Monge-Kantorovich transportation steps on Gaussian distributions and studying the induced holonomy of the optimal transportation problem. From this vantage point we determine the minimal number of positive-definite factors that have a desired effect on the spectrum of the product, e.g., ensure specified eigenvalues or being a rotation matrix. Our approach is computational and allows to identify the needed number of factors as well as trade off their conditioning number with their actual number.