The Pseudoinverse of A=CR is A^+=R^+C^+ (?)

05/02/2023

∙

by   Michał P. Karpowicz, et al.

∙

0

∙

share

The statement in the title is not generally true, unless C and R have full rank. Then the m by r matrix C is assumed to have r independent columns (rank r). The r by n matrix R is assumed to have r independent rows (rank r). In this case the pseudoinverse C^+ is the left inverse of C, and the pseudoinverse R^+ is the right inverse of R. The simplest proof of A^+ = R^+C^+ verifies the four Penrose identities that determine the pseudoinverse A^+ of any matrix A. Our goal is a different proof of A^+ = R^+C^+, starting from first principles. We begin with the four fundamental subspaces associated with any m by n matrix A of rank r. Those are the column space and nullspace of A and A^T. The proof of A^+ = R^+C^+ then shows that this matrix acts correctly on every vector in the column space of A and on every vector in the nullspace of A^T.