Orthant–Monotonic Norms and Overdetermined Linear Systems Original Research Article

In the paper the properties of ‖·‖-approximate solutions of real overdetermined linear systemsAx=bare investigated. We characterize the norms for which the approximate solutions are in the convex hull of the basic solutions ofAx=b. For this purpose we prove some properties of the orthant–monotonic norms. We derive an explicit formula which expresses the approximate solutions ofAx=b, with respect to some norms, as convex combinations of the basic solutions for ann+1 bynreal matrixAof rank n. Moreover, we consider the relation between the sets of the weighted approximate solutions for the orthant–monotonic norms satisfying some additional conditions.