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.