Principal pivot transforms are maps between matrices that generalize, in some sense, Schur complementation and inversion. They are a somewhat counterintuitive but powerful tool that allows one to
reinterpret Gaussian elimination-type algorithms in a different framework.
We will show how they can deal successfully with various kinds of matrices in a structure-preserving way; this includes (weakly) quasi-definite matrices, i.e., symmetric matrices with two complementary
diagonal blocks of opposite definiteness, as well as Hamiltonian/symplectic matrices. We will then present an algorithm to solve dense algebraic Riccati equations in a factored form, which relies on these transforms as a basic building block.