A characterization for the W-weighted Drazin inverse and a Cramer rule for the W-weighted Drazin inverse solution

Abstract

We establish characterization for the W-weighted Drazin inverse of an arbitrary rectangular matrix which reduces to the well-known result if the matrix is nonsingular. Also, a Cramer rule for finding the unique W-weighted Drazin inverse solution x∈ R[( AW) k 1 ] of special restricted linear equations WAWx=b, b∈R[(WA) k 2 ] is presented, and reduces to the classical Cramer rule if A is invertible.