Matrices with special structures arise in numerous applications. In some cases, such as quasidefinite matrices or their generalizations, we can exploit this special structure. If the matrix H is quasidefinite, we propose a new variant of the symmetric indefinite factorization. We show that linear system Hz= b, H quasidefinite with a special structure, can be interpreted as an equilibrium system. So, even if some blocks in H are ill–conditioned, the important part of solution vector z can be accurately computed. In the case of a generalized quasidefinite matrix, we derive bounds on number of its positive and negative eigenvalues.