Unifying unitary and hyperbolic transformations

Abstract

In this paper, we describe unified formulas for unitary and hyperbolic reflections and rotations, and show how these unified transformations can be used to compute a Hermitian triangular decomposition R ̂ H D R ̂ of a strongly nonsingular indefinite matrix A ̂ given in the form A ̂ =X 1 H X 1+αX 2 H X 2, α=±1 . The unification 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 unified transformations, propose pivoting strategies for lowering the condition number of the transformations, and present a unified stability analysis for applying the transformations to a matrix.