Matrix completion and semidefinite matrices

TL;DR

This paper extends the theory of matrix completion from positive definite matrices to positive semidefinite matrices, introducing generalized determinants and inverses to handle singular cases.

Contribution

It generalizes the uniqueness results for matrix completion to semidefinite matrices, including singular cases of maximal rank, using new mathematical tools.

Findings

Generalized determinant for semidefinite matrices introduced

Unique completion characterized via generalized inverses

Results applicable to matrices of maximal rank

Abstract

Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes the determinant, or as the matrix whose inverse has zeroes in those places that were undetermined in the original matrix. This paper extends these uniqueness results to the case of semidefinite matrices. Because the determinant vanishes for singular matrices, and because the inverse does not exist, we introduce a generalized determinant and use generalized inverses to formulate equivalent characterizations in the semidefinite case. For a class of matrices that are singular but of maximal rank unique characterizations can be given, just as in the positive definite case.