Vanishing minor conditions for inverse zero patterns
The authors are interested in the following kind of question: Suppose that certain entries of a matrix \(A\) are specified to be zero; what can we deduce from this information alone about vanishing of determinantal minors of \(A\) and \(A^{-1}\)? They term such minors as ``generically'' zero. Their starting point is a classical theorem due to Frobenius (1912): if \(A\) is a \(k\times k\) matrix, then \(A\) has an \(r\times s\) submatrix of zero entries with \(r+s>k\) if and only if \(\text{det }A\) is generically 0. The paper introduces a lot of notation, and the results obtained are rather technical, but we paraphrase one of them (Theorem 4) to give the flavour of the paper. Let \(A\) be an invertible \(n\times n\) matrix, and partition \(\{1,\dots,n\}\) into three nonempty subsets \(\alpha\), \(\beta\) and \(\gamma\). Suppose that the submatrix \(A[\beta\mid\gamma]\) whose rows and columns are indexed by \(\beta\) and \(\gamma\), respectively, has all entries equal to 0. Then for all \(i\in \beta\) and \(j\in\gamma\), we have \(\text{det }A^{-1} [\alpha\cup \{i\}\mid\alpha\cup \{j\}]=0\).
scientific article; zbMATH DE number 166231