The geometry of basic, approximate, and minimum norm solutions of linear equations
نویسنده
چکیده
The basic solutions of the linear equation 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). The residual r(x) of a vector x is r := Ax − b. Given 1 ≤ p ≤ ∞, the lp-approximate solutions of Ax = b, denoted x , are minimizers of ‖r(x)‖p. Given M ∈ Dm, the set of positive diagonal m×m matrices, the solutions of min x ‖M (Ax− b)‖p , are called scaled lp-approximate solutions. For 1 ≤ p1, p2 ≤ ∞, the minimum lp2-norm lp1-approximate solutions are denoted x {p1} {p2} . Main results: (a) The set of scaled lp-approximate solutions, with M ranging over Dm, is the same for all 1 < p < ∞. (b) If A ∈ IR m , C contains all [some] minimum lp-norm solutions, for 1 ≤ p < ∞ [p = ∞]. (c) For general A, 1 ≤ p1, p2 < ∞, the set C contains all x {p1} {p2} .
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