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

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 equationsWAWx=b,b@?R[(WA)^k^"^2]is presented, and reduces to the classical Cramer rule if A is invertible.

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