On the Factorization of Hyperbolic and Unitary Transformations into Rotations

This paper presents a Σ-unitary analogue to the CS decomposition of a partitioned unitary matrix. The hyperbolic rotations revealed by the decomposition are shown to be optimal in that, among a broader class of decompositions of Σ-unitary matrices into elementary hyperbolic rotations, they are the smallest possible in a sum-of-squares sense. A similar optimality property is shown to hold for the sines in the CS decomposition of a unitary matrix.