The authors consider the preconditioning of a matrix that has the form with symmetric positive definite of size and symmetric of size and a nonsingular Schur complement with . In the literature one usually deals with a saddle point context where is negative definite. The preconditioner considered is exactly like , but is replaced by a symmetric positive definite approximant , hence the name constrained preconditioner.
First, a positive and a negative interval are found that contain the eigenvalues of the symmetric matrix which depends on the locations of the intervals containing the eigenvalues of , , and . Next the spectrum of is investigated, mainly dealing with the multiplicity of the eigenvalue 1, but also with the structure of the eigenvectors. These properties depend on the (dimension of the) null space of . Numerical examples illustrate the method.