The Real Positive Definite Matrix Completion Problem: An Optimization Viewpoint

Abstract

We look at the real positive (semi)definite matrix completion problem from the convex optimization viewpoint.

The problem is introduced via relative entropy minimization, transformed into the standard max-det from, and conditions are sought for existence of positive definite and positive semidefinite completions. Using basic tools of convex optimization, a unifying view of the existence and uniqueness problem for positive (semi)definite matrix completions is presented. Some results previously established using functional-analytic techniques are recovered and some new are given.

In particular, the maximum determinant completion is generalized to the positive semidefinte matrices with an arbitrary sparsity pattern.

Keywords: positive (semi) definite matrix, matrix completion

JEL Classification: C61