On the shape of the symmetric, persymmetric, and skew-symmetric solution set

We present a characterization of the solution set S, the symmetric solution set S-sym, the persymmetric solution set S-per, and the skew-symmetric solution set S-skew of real linear systems Ax = b with the n x n coefficient matrix A varying between a lower bound A and an upper bound (A) over bar, and with b similarly varying between b, (b) over bar. We show that in each orthant the sets S-sym, S-per, and S-skew are, respectively, the intersection of S with sets, the boundaries of which are quadrics.