Partial matrices whose completions have ranks bounded below
Let \(A\) be an \(m\times n\) partial matrix over a field~\(\mathbb{F}\). That is, its some entries are specified while the other entries are indeterminates, and specifying them produces a completion of~\(A\). Assuming that \(m=n\) and that \(\mathbb{F}\) has at least \(n+1\) elements, \textit{R.~A.~Brualdi}, \textit{Z.~Huang} and \textit{X.~Zhan} [Linear Algebra Appl.~433, 1452--1462 (2010; Zbl 1205.15042)] proved that if all the completions of~\(A\) have rank~\(n\), then the number of indeterminates in~\(A\) is at most \(\frac{1}{2}n(n-1)\). They also described the case when this bound is attained. The present authors pursue this topic further. Let \(k\leq m,n\) and assume that every completion of~\(A\) has rank at least~\(k\). The authors prove that the number of indeterminates in~\(A\) is at most \(mn-\frac{1}{2}k(k+1)\). They also describe the case when this bound is attained. Their main theoretical tool is a duality relationship between certain affine spaces of matrices.
scientific article; zbMATH DE number 5924587