Inertia properties of indefinite quadratic forms

We study the relation between the solutions of two estimation problems with indefinite quadratic forms. We show that a complete link between both solutions can be established by invoking a fundamental set of inertia conditions. While these inertia conditions are automatically satisfied in a standard Hilbert space setting, they nevertheless turn out to mark the differences between the two estimation problems in indefinite metric spaces. They also include, as special cases, the well-known conditions for the existence of H/sup -/spl infin//-filters and controllers. Given two Hermitian matrices {/spl Pi/, W}, a column vector y, and an arbitrary matrix A of appropriate dimensions, we study the relation between two minimization problems with quadratic cost functions, and also refer to the indefinite-weighted least-squares problem.