Schur complements obey Lambek's categorial grammar: Another view of Gaussian elimination and LU decomposition

Abstract

For three decades Schur complements have seen increasing applications in linear algebra, often as abstractions of Gaussian elimination. It is known that they obey certain nontrivial identities, such as Crabtree and Haynsworth's quotient property. We began this work asking if there were a theory for deciding their properties in general. Lambek's Categorial Grammar is a deductive system formalized in 1958 by Lambek as a mathematical foundation for a syntactic calculus of language. We show that Categorial Grammar gives a deductive system for deriving identities obeyed by LU-and UL-decompositions, Gaussian elimination, and Schur complements. At first impression this seems to be a strange result, connecting two unrelated topics. In retrospect, though, it is a consequence of the way both use quotients. It may have applications in developing grammatical formalisms and numerical algorithms.