Inherited Matrix Entries: Principal Submatrices of the Inverse

Abstract

For a nonsingular n-by-n matrix $A = [a_{ij} ]$, let $\alpha \subseteq \{ 1,2, \cdots ,n\} $ and let $A[\alpha ]$ denote the principal submatrix of A lying in the rows and columns indicated by $\alpha $. We determine the combinatorial circumstances under which the $(i,j)$ entry of the Schur complement $(A^{ - 1} [\alpha ])^{ - 1} $ equals $a_{ij} $, and under which the graph of this Schur complement is contained in the graph of $A[\alpha ]$.