The set of real *positive definite matrices* of dimension *n* is a subset *P*_{n} of all matrices *R*^{n × n}.^{1} This subset is especially relevant in Statistics as covariance matrices are real positive definite, but some statistical computation can be difficult because *P*_{n} 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 *P*_{n} to the R software system.

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

See Positive-definite matrices or Rajendra Bhatia

*Positive Definite Matrices*Chapter 1↩P.-A. Absil, R. Mahony, and R. Sepulchre

*Optimization Algorithms on Matrix Manifolds*Princeton University Press 2008 full text on line↩