On ridge operators

Let X be an n by p matrix, and define RX(λ)=X(X′X+λPX′)-X′, which is called a ridge operator, where λ is a nonnegative constant (called the ridge parameter), and PX′=X′(XX′)-X. Various properties of RX(λ) were discussed, including additive decompositions of this matrix similar to those of PX≡RX(0)=X(X′X)-X′, the orthogonal projector onto the range space of X. These properties and decompositions are useful, especially in ridge estimation of reduced rank regression and multiple-set canonical correlation analyses.