Least-square solutions of inverse problems for reflexive matrices

P = (p_ij) ∈ C^m×n is regarded as a generalized reflection matrix if P satisfies that P^H = P, P² = I. Let P ∈ C^m×n be a given generalized reflection matrix, A matrix A ∈ C^m×n is regarded as an n × n reflexive matrix with respect to P if A satisfies A = PAP. We denote the set of all n × n reflexive matrices by C_r^m×n(P). In this paper, the least-square solutions of the inverse problem of reflexive matrices is discussed, and the expression of the solution is obtained. In addition, the problem of using reflexive matrices to construct the optimal approximation to a given matrix is discussed, the necessary and sufficient conditions about the problem are derived, and the expression of the solutions is provided.