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Abstract: In this paper, we proposed an algorithm for solving the linear systems of matrix equations over the generalized -reflexive matrix (). According to the algorithm, the solvability of the problem can be determined automatically. When the problem is consistent over the generalized -reflexive matrix , for any generalized -reflexive initial iterative matrices , the generalized -reflexive solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized -reflexive solution can also be derived when the appropriate initial iterative matrices are chosen. A sufficient and necessary condition for which the linear systems of matrix equations is inconsistent is given. Furthermore, the optimal approximate solution for a group of given matrices can be derived by finding the least-norm generalized -reflexive solution of a new corresponding linear system of matrix equations. Finally, we present a numerical example to verify the theoretical results of this paper. [Copyright &y& Elsevier]

Aim: “Natural” HLA antibodies are reported in non-alloimmunized healthy males, but these antibodies are normally not against self-HLA antigens. We report a C∗04:01 antibody against both self and donor in a 47 year-old African American female who received a renal transplant. Methods: Class I and class II PRA and specificity were performed by Labscreen PRA beads and Single Antigen beads (SAB) (One Lambda). The serum was further tested with acid-treated (denatured antigen) SAB and reduced-amount denatured antigen beads (iBeads, One Lambda) and C1qScreen beads (One Lambda). Results: Pre-transplant sequential sera showed the presence of weak C∗04:01 antibody (MFI∼900) though she was Cw4-positive (as determined by both DNA and cytotoxic methods). She received a kidney transplant from a 5/10 HLA mismatched C∗04-positive deceased donor with negative flow cytometric and cytotoxicity allo-crossmatches. At 2.5 months post-transplant, her serum creatinine increased to 2.7mg/dL and the C∗04:01 antibody MFI increased to 2875. To determine whether the C∗04:01 antibody was an allelic donor specific antibody, the patient and donor were HLA typed by high-resolution methods and it was confirmed that both carried the HLA-C∗04:01 allele. The MFI of the C∗04:01 antibody decreased along with titration of the serum, but did not change with DTT treatment. The C∗04:01 antibody was also detected with acid-treated beads, but not detected with iBeads. Furthermore, this antibody did not show complement-binding activity by C1qScreen. No antibody mediated rejection/C4d deposition was observed in her graft biopsy. She was successfully treated for acute drug toxicity. Conclusions: To our knowledge, this is the first report of the presence of a self-allelic HLA antibody detected in a kidney transplant patient when both donor and recipient carry the same allele. It may be an antibody against cryptic epitopes. Importantly, it does not show complement-binding capability, thus, it may not have a detrimental effect to the kidney graft. [Copyright &y& Elsevier]