Answer to Question #24836 in Linear Algebra for Jacob Milne

Question

Associated with each equation (Ax)_i=b_i in the system Ax=b there is a hyperplane H_i defined to be the subset of J-dimensional column vectors given by: H_i = {x|(Ax)_i = b_i}.

Show that, for any vector z in C^J, the member of H_i closest to z is x having the entries

x_j = z_j + a_i^(-1)[Aijconjugate](b_i - (Az)_i),

where

a_i = the sum from j=1 to J |Aij|^2.