Finding an Optimal Proximity Bound in a Very Special Scenario

Abstract

In order to introduce the main result and also provide some background material, we require some notation. Let A = (a ij ) ∈ Z m×n with m < n and let τ = {i 1 , . . . , i k } ⊂ {1, . . . , n} with i 1 < • • • < i k be an index set. We will use the notation A τ for the m × k submatrix of A with columns indexed by τ . In a similar manner, given x ∈ R n , we will denote by x τ the vector (x i1 , . . . , x i k ) T . The complement of τ in {1, . . . , n} will be denoted by τ = {1, . . . , n}\τ . We will say that τ is a basis if |τ | = m and the submatrix A τ is nonsingular. In the scenario that τ is a basis, we will replace τ by γ. Further, denote by ∆(A) the maximum absolute valued m-dimensional subdeterminant of A, namely