Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations and variational problems. In these applications one is often interested in computing a factorization that puts into evidence the inertia of the matrix or possibly provides an estimate of its eigenvalues. In this talk we
present an algorithm that provides this information for any symmetric indefinite matrix by transforming it to a block anti-triangular form using orthogonal similarity transformations. We show that the algorithm is backward stable and that it has a complexity that is comparable to existing matrix decompositions for dense indefinite matrices.