Using the GSVD and the lifting technique to find {P,k+1}{P,k+1} reflexive and anti-reflexive solutions of AXB=CAXB=C

The generalized singular value decomposition (GSVD) and the lifting technique combined with the Kronecker product are exploited to find reflexive and anti-reflexive (with respect to a generalized {k+1}{k+1}-reflection matrix PP) solutions of the matrix equation AXB=CAXB=C. The computational cost of the presented algorithm is studied and several numerical examples are presented.