Rank Equalities Related to the Generalized Inverses <i>A</i>^{‖(<i>B</i>₁,<i>C</i>₁)}, <i>D</i>^{‖(<i>B</i>₂,<i>C</i>₂)} of Two Matrices <i>A</i> and <i>D</i>

Abstract

Let A be an n × n complex matrix. The ( B , C ) -inverse A ∥ ( B , C ) of A was introduced by Drazin in 2012. For given matrices A and B, several rank equalities related to A ∥ ( B 1 , C 1 ) and B ∥ ( B 2 , C 2 ) of A and B are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained.

Keywords

• Rank
• (<i>B</i>, <i>C</i>)-inverse
• inverse along an element