A Nonnegativity Criterion for Erdős Matrices

Let A be an n × n n× n n times n doubly stochastic matrix. The Marcus–Ree inequality asserts that ∥ A ∥ F 2 ≤ maxtrace ( A ) \|A\|F²≤ maxtrace(A) StartMetric upper A EndMetric Subscript normal upper F Superscript 2 Baseline less than or equals maxtrace left parenthesis upper A right parenthesis . Matrices attaining equality are called Erdős matrices. Recent work of Karmakar et al. [‘Characterization of Erdős matrices by their zero entries’, Linear Algebra Appl. 739 (2026), 154–169] shows that every Erdős matrix is a restricted common diagonal sum matrix. By the structural theory of Brualdi and Dahl [‘Diagonal sums of doubly stochastic matrices’, Linear Multilinear Algebra 70 (2022), 4946–4972], a restricted common diagonal sum matrix with fully indecomposable skeleton S = ( s i j ) S=(sᵢⱼ) upper S equals left parenthesis s Subscript i j Baseline right parenthesis admits additive potentials satisfying a i j = ( u i + v j ) s i j aᵢⱼ=(uᵢ+vⱼ)sᵢⱼ a Subscript i j Baseline equals left parenthesis u Subscript i Baseline plus v Subscript j Baseline right parenthesis s Subscript i j . Karmakar et al. observed that min i u i + min j v j ≥ 0 min ᵢ uᵢ+min ⱼvⱼ≥ 0 min Underscript i Endscripts u Subscript i Baseline plus min Underscript j Endscripts v Subscript j Baseline greater than or equals 0