On the product of oblique projectors

Abstract

In this paper we show that for two projectors (idempotents) P ₁ and P ₂, idempotency of the product P ₁ P ₂ is equivalent to the coincidence of the range of P ₁ P ₂ and a certain subspace, which only depends on the onto-and along-spaces of P ₁ and P ₂. Some further investigations are made, and necessary and sufficient conditions for the commutativity of two projectors are given.

Keywords:

• Oblique projector
• idempotent matrix
• product of oblique projectors
• commut-ativity