On the Shape of the Symmetric, Persymmetric, and Skew-Symmetric Solution Set

Abstract

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 $ A x = b $ with the $ n \times n $ coefficient matrix $ A $ varying between a lower bound $ \ul A $ and an upper bound $ \ol A $, and with $ b $ similarly varying between $ \ul b,\ \ol b $. 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.