Maximum determinant positive definite Toeplitz completions

TL;DR

This paper characterizes when the maximum determinant positive definite Toeplitz completion of a partial matrix is itself Toeplitz, extending to semidefinite cases and linking to optimization problem singularity.

Contribution

It provides a characterization of patterns for Toeplitz completions with maximum determinant and extends results to semidefinite and rank maximization scenarios.

Findings

Identifies patterns where maximum determinant completion is Toeplitz.

Extends results to positive semidefinite matrices.

Links completion properties to singularity degree in optimization.

Abstract

We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite replaced by positive semidefinite, and maximum determinant replaced by maximum rank. These results are used to determine the singularity degree of a family of semidefinite optimization problems.