HISTORY AND GENERALITY OF THE CS DECOMPOSITION

It is almost a quarter of a century since Chandler Davis and William K ahan brought together the key ideas of what Stewart later completed an d defined to be the CS decomposition (CSD) of a partitioned unitary ma trix. This paper outlines some germane points in the history of the CS D, pointing out the contributions of Jordan, of Davis and Kahan, and o f Stewart, and the relationship of the CSD to the ''direct rotation'' of Davis and Kato. The paper provides an easy to memorize, constructiv e 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 dec ompositions, and points out some useful nullity properties one form of the CSD trivially reveals. Finally it shows how, via the QR factoriza tion, the CSD can be used to obtain interesting results for partitione d nonsingular matrices. We suggest the CSD be taught in its most gener al form with no restrictions on the two by two partition, and initiall y with no mention of angles between subspaces.