Complementary Schur complements
Let \(M\) and \(N\) be \(m\times m\) and \(n\times n\) complex matrices, respectively. The authors define \(M\) and \(N\) to be ``Schur coupled'' if there exist matrices \(A,B,C\) and \(D\) such that \(M=D-CA^{-1}B\) and \(N=A- BD^{-1}C\). The main result of this paper is that the following are equivalent: (i) \(M\) and \(N\) are Schur coupled; (ii) there is an \((m+n)\times(m+n)\) matrix \(H\) such that the leading \(m\times m\) principal submatrix of \(H\) is \(M\) and the final \(n\times n\) principal submatrix of \(H^{-1}\) is \(N\); and (iii) \(m-\text{rank} M=n-\text{rank} N\).
scientific article; zbMATH DE number 553979