If G(C) denotes the minimal Gersgorin set for C ∈ Cn,n, and if, for any nonempty subset α of the first n positive integers, C[α] denotes the principal minor of C determined by α, then conditions are determined which characterize matrices A and B in Cn,n such that the inclusions G((D + B)[α]) ⊆G((D + A)[α]) are valid for all subsets α of the first n positive integers, and for all diagonal matrices D in Cn,n. Connections with the newly defined set of ω-matrices are also included.
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