History and generality of the CS decomposition

AbstractIt is almost a quarter of a century since Chandler Davis and William Kahan brought together the key ideas of what Stewart later completed and defined to be the CS decomposition (CSD) of a partitioned unitary matrix. This paper outlines some germane points in the history of the CSD, pointing out the contributions of Jordan, of Davis and Kahan, and of Stewart, and the relationship of the CSD to the “direct rotation” of Davis and Kato. The paper provides an easy to memorize, constructive proof of the CSD, reviews one of its important uses, and suggests a motivation for the CSD which emphasizes how generally useful it is. It shows the relation between the CSD and generalized singular value decompositions, and points out some useful nullity properties one form of the CSD trivially reveals. Finally it shows how, via the QR factorization, the CSD can be used to obtain interesting results for partitioned nonsingular matrices. We suggest the CSD be taught in its most general form with no restrictions on the two by two partition, and initially with no mention of angles between subspaces