The solvability conditions of matrix equations with k-involution

Let m × m complex matrix P and n × n complex matrix Q be k-involutions, i.e., Pk−1 = P, Qk−1 = Q for some integer k ≥ 2. An m × n complex matrix A is (P, Q, β)symmetric if PAQ = λβA, or (P, Q, α, β)-symmetric if PAQ−α = λβA, where λ = e2πi/k and α, β ∈ {1, 2, . . . , k}. In this paper, for given matrices X, Y, E, F with appropriate sizes, the solvability of matrix equations AX = E and Y A = F under (P, Q, β)and (P, Q, α, β)-constraints, respectively, are investigated. Meanwhile, the associated optimal approximation problem is also considered when the above P and Q are unitary.