On the Numerical Analysis of Oblique Projectors

An oblique projector is an idempotent matrix whose null space is oblique to its range, in contrast to an orthogonal projector, whose null space is orthogonal to its range. Oblique projectors arise naturally in many applications and have a substantial literature. Missing from that literature, however, are systematic expositions of their numerical properties, including their perturbation theory, their various representations, their behavior in the presence of rounding error, the computation of complementary projections, and updating algorithms. This article is intended to make a start at filling this gap. The first part of the article is devoted to the first four of the above topics, with particular attention given to complementation. In the second part, stable algorithms are derived for updating an XQRY representation of projectors, which was introduced in the first part.