Abstract: Let P ∈ C^n×n be a Hermitian and {k + 1}-potent matrix, i.e., P^k+1= P = P^*,where（·）^*stands for the conjugate transpose of a matrix. A matrix X ∈ Cn×nis called{P, k + 1}-reflexive（anti-reflexive） if PXP = X（P XP =-X）. The system of matrix equations AX = C, XB = D subject to {P, k + 1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases: k = 1 and k = 2, the least squares solution and the associated optimal approximation problem are also considered.

Key words: system of matrix equations, potent matrix, {P, k+1}-reflexive (anti-reflexive), approximation problem, least squares solution