Positive definite matrices arise in several contexts: quantum mechanics, statistics, machine learning, image processing, elasticity etc. Various notions of averaging are useful in different contexts. In the 1970’s physicists, electrical engineers, and matrix theorists developed a notion of a geometric mean of two positive definite matrices. The problem of extending the concept to more than two matrices remained open for several years. The problem was resolved in 2004 using ideas from Riemannian geometry. Meanwhile, effective use of the idea has been made in applications in radar data processing and imaging of the brain. In this talk, I will explain the main ideas from the perspective of matrix analysis.
Reference: R. Bhatia, Positive Definite Matrices, Princeton University Press 2007, Hindustan Book Agency 2007.