Unifying Unitary and Hyperbolic Transformations

In this paper we describe uniied formulas for unitary and hyperbolic reeections and rotations, and show how these uniied transformations can be used to compute a Hermitian triangular decomposition ^ R H D ^ R of a strongly nonsingular indeenite matrix ^ A given in the form ^ The uniication is achieved by the introduction of signature matrices which determine whether the applicable transformations are unitary, hyperbolic, or their generalizations. We derive formulas for the condition numbers of the uniied transformations, propose pivoting strategies for lowering the condition number of the transformations, and present a uniied stability analysis for applying the transformations to a matrix.