Explicit Expressions of the Generalized Inverses and Condensed Cramer Rules

Authors

Jun Ji, Kennesaw State UniversityFollow

Department

Mathematics

Document Type

Article

Publication Date

7-15-2005

Abstract

In this paper, we obtain an explicit representation of the {2}-inverse A_T,S⁽²⁾ of a matrix A ∈ C^m×n with the prescribed range T and null space S. As special cases, new expressions for the Moore-Penrose inverse A⁺ and Drazin inverse A^D are derived. Through explicit expressions, we re-derive the condensed Cramer rules of Werner for minimal-norm least squares solution of linear equations Ax = b and propose two new condensed Cramer rules for the unique solution of a class of singular system Ax = b, x ∈ R(A^k), b ∈ R(A^k), k = Ind(A). Finally, condensed determinantal expressions for A⁺, A ^D, AA⁺, A⁺A, and AA^D are also presented.