The geometry of basic, approximate, and minimum-norm solutions of linear equations

Abstract

The basic solutions of the linear equations Ax = b are the solutions of subsystems corresponding to maximal nonsingular submatrices of A. The convex hull of the basic solutions is denoted by C = C( A, b). Given 1 ≤ p ≤ ∞, the l p - approximate solutions of Ax = b, denoted x { p} , are minimizers of ∥ Ax − b∥ p . Given M ∈ D m , the set of positive diagonal m × m matrices, the solutions of min x ∥ M( Ax − b)∥ p are called scaled l p - approximate solutions. For 1 ≤ p 1, p 2 ≤ ∞, the minimum- l p2 -norm l p1 - approximate solutions are denoted x {p 1} {p 2} . Main results: 1. (1) If A ∈ R m × n m , then C contains all [some] minimum l p -norm solutions, for 1 ≤ p < ∞ [ p = ∞]. 2. (2) For general A and any 1 ≤ p 1, p 2 < ∞ the set C contains all x {p1} {p2}. 3. (3) The set of scaled l p -approximate solutions, with M ranging over D m , is the same for all 1 < p < ∞. 4. (4) The set of scaled least-squares solutions has the same closure as the set of solutions of min x f (| Ax − b|), where f:R m + → R ranges over all strictly isotone functions.