On completions of symmetric and antisymmetric block diagonal partial matrices

TL;DR

This paper investigates the maximum and minimum ranks achievable through completions of symmetric and antisymmetric block diagonal partial matrices, focusing on cases where only diagonal blocks are specified.

Contribution

It provides explicit formulas for the maximum rank of symmetric completions and the minimum and maximum ranks of antisymmetric completions with given diagonal blocks.

Findings

Maximum rank of symmetric completions determined

Minimum and maximum ranks of antisymmetric completions established

Results applicable to partial matrices with specified diagonal blocks

Abstract

A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.