Abstract: Presents an iterative algorithm to compute the optimal approximation solutions of the generalized Sylvester matrix equations AXB + CX^TD = E over reflexive（anti-reflexive）matrices with the hybrid steepest descent method. Whether the matrix equations AXB + CX^TD = E are consistent or not，for arbitrary initial reflexive（anti-reflexive）matrix X₀ ，the given algorithm can be used to compute the reflexive（anti-reflexive）optimal approximation solutions X . The effectiveness of the proposed algorithm is verified by two numerical examples.

Key words: Sylvester matrix equations, Kronecker product, hybrid steepest descent method, optimal approximation, reflexive matrix