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 × 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 ≤ r. The active submatrix at each step of the Gaussian elimination applied to A is the Schur complement of the appropriate leading principal submatrix. Bibliography: 2 titles.