Typical ranks in symmetric matrix completion

TL;DR

This paper investigates the typical ranks in symmetric matrix completion, providing combinatorial characterizations of patterns with specific typical ranks and conditions for minimal completion to rank n.

Contribution

It offers a combinatorial framework to determine typical ranks and minimal completability for symmetric matrices with specified entries, especially when the entries are real.

Findings

Patterns with n as a typical rank are characterized combinatorially.

Conditions for minimal completion to rank n are established.

Typical ranks for patterns with low maximal typical rank are described.

Abstract

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks. We give a combinatorial description of the patterns of specified entires of $n\times n$ symmetric matrices that have $n$ as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank $n$. We also characterize the typical ranks for patterns of entries with low maximal typical rank.