Let xprtnbspbe the generalized inverse of a n x p matrix x

Question

Let X⁺=P?⁺_(r)^T be the generalized inverse of a (n x p) matrix X (Theorem 5.1). Show that the following relations hold:

THEOREM 5.1

(The Singular Value Decomposition Theorem) Let X be a (n x p) data matrix of reduced rank rpthen there exists a real (n × r) matrix Q, a real ( p × r ) matrix P, and a real (r × r) diagonal matrix

?_(r) such that