Let $M$ be a (nonsingular) partitioned matrix $$ M= \left( \matrix A & B^T \ B & 0 \ \endmatrix \right), $$ where $B$ is $m\times n$ ($m\leq n$) of full rank, and $A$ is symmetric, indefinite, and positive definite on the kernel of $B$. The authors provide estimates for real intervals containing the eigenvalues of $M$. Some refinements on the eigenvalue bounds are given by including additional knowledge of the data. An equivalent augmented formulation of the original problem is considered. The authors also study the so-called “stabilized” saddle point matrix $$M_C= \left( \matrix A & B^T \ B & -C \ \endmatrix \right),$$ where $C$ is symmetric and positive semidefinite. Interval estimates for the spectrum of $M_C$ are provided. At the end, natural generalizations of a certain class of preconditioners are discussed.
Reviewer: C. M. da Fonseca (Coimbra)