Gaussian elimination and the ranks of the components in the Cartesian decomposition of a matrix

Let $A=B+iC$, where $B=B^*$, $C=C^*$, be the Cartesian decomposition of an $n\times n$ matrix $A$, and let the component $B$ (or $C$) have rank $r<n$. It is shown that for a nonsingular $A$, the inverse $A^{-1}$ has an analogous property. This implies that all the (correctly defined) Schur complements in $A$ have Cartesian decompositions with component $B$ (or $C$) of rank $\le r$. The active submatrix at each step of the Gaussian elimination applied to $A$ is the Schur complement of the appropriate leading principal submatrix. Bibl. – 2 titles.