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Several full-rank representations of the $A_{T,S}^{(2)}$ inverse of a given constant complex matrix, which are based on various complete orthogonal factorizations, are introduced. Particularly, we introduce a full rank representation based on the Singular Value Decomposition ($SVD$) as well as on a combination of the $QR$ decomposition and the $SVD$ of the triangular matrix produced by the $QR$ decomposition. Furthermore, representations based on the factorizations to a bidiagonal form are defined. The representations arising from reductions to bidiagonal form are applicable to real full row rank matrices. Illustrative numerical examples as well as an extensive numerical study are presented. A comparison of three introduced methods is presented.

Computing Outer Inverses Using Complete Orthogonal Factorizations