The Shape of the Symmetric Solution Set

Abstract

We give a new deduction of the set of inequalities which characterize the solution set S of real linear systems Ax = b with the n × n coefficient matrix A varying between a lower bound \(\underline A \) and an upper bound \(\overline A \), and with b similarly varying between \(\underline b \) and \(\overline b \). The idea of this deduction can also be used to construct a set of inequalities which describe the so-called symmetric solution set S sym, i.e., the solution set of Ax = b with A = A T varying between the bounds \(\underline A = {\underline A ^T}\) and \(\overline A = {\overline A ^T}.\) This is the main result of our paper. We show that in each orthant S sym is the intersection of S with sets of which the boundaries are quadrics.