Restoring rank and consistency by orthogonal projection

Let A0x=b0 be a consistent (but possibly unknown) linear algebraic system of m equations in n unknowns, with rank(A0)=k (likewise possibly unknown). Let Ax=b be a known (but possibly inconsistent) nearby system. Procedures for "solving" Ax=b usually replace it (at least in principle) by a nearby consistent system A ̃x=b ̃ of (hopefully) rank k, and solve that one instead. We consider consistent systems A ̃x=b ̃ with rank(A ̃)=k such that (Ã,b̃) is the orthogonal projection of (A,b) on span(Ã) [i.e., the columns of (Ã{dashed fence}b̃) are the projections of those of (A {dashed fence} b)]. Under suitable circumstances the rank k pair (Ã,b̃) nearest to (A,b) in the sense of minimizing {norm of matrix}(A ̃{dashed fence}b ̃) - (A ̃{dashed fence}b ̃){norm of matrix}F belongs to this class, as well as the pair delivered by the ordinary least squares method (if k=n≤m) or, more generally, the pair delivered by a well-known algorithm (viz. HFTI). If ε{lunate}:= |(A{dashed fence}b)-(A0{dashed fence}b0){norm of matrix}F, then the minimum length solutions of all systems A ̃x=b ̃ so related to (A,b) are shown to differ mutually only by O(ε{lunate}2). This means, e.g., that itwill usually not pay to compute the solution of the nearest system.