Student project posted Feb 3, 2015 (

An R suite for the geometry of positive definite matrices

The set of real positive definite matrices of dimension n is a subset Pn of all matrices Rn × n.1 This subset is especially relevant in Statistics as covariance matrices are real positive definite, but some statistical computation can be difficult because Pn is not a linear space. For example, in some relevant applications the data consists of empirical covariance matrices. Some very basic operation, such as computing the mean are to be defined carefully.

The set of positive definite matrices is an open subset of the Euclidean space of symmetric matrices, that is, it has a Riemannian geometrical structure that is usable to solve optimization problems with analytic methods.2 A very well documented collection of MatLab programs is available.3

It is interesting and potentially useful to port the optimization algorithms for Pn to the R software system.

This project requires some background in matrix algebra, differential manifolds, Riemannian geometry, Statistics, MatLab and R programming.

Student project to be posted 2015 (

  1. See Positive-definite matrices or Rajendra Bhatia Positive Definite Matrices Chapter 1

  2. P.-A. Absil, R. Mahony, and R. Sepulchre Optimization Algorithms on Matrix Manifolds Princeton University Press 2008 full text on line

  3. MatLab suite manopt.