Some comments about measures, Jacobians and Moore-Penrose inverse

TL;DR

This paper discusses Jacobian computations for matrices, focusing on the Moore-Penrose inverse and deriving Jacobians in full rank cases using exterior product theory, clarifying existing methods.

Contribution

It revisits the Jacobian of the Moore-Penrose inverse and derives Jacobians for full rank matrices using classical exterior product theory.

Findings

Revised Jacobian of Moore-Penrose inverse via differential calculus

Derived Jacobian for full rank matrices using exterior product

Clarified relationships between Jacobians and matrix rank

Abstract

Some general problems of Jacobian computations in non-full rank matrices are discussed in this work. In particular, the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is revisited. Then the Jacobian in the full rank case is derived under the simple and old theory of the exterior product.