Generalized inverses and solutions of linear systems
John Z. Hearon · Journal of Research of the National Bureau of Standards Section B Mathematical Sciences · 1968
For an arbitrary complex matrix A we consider (1) th e set of all matrices B such that ABA =A and AB is H ermitian and (2) the set of all matri ces B s uc h that ABA = A and BA is Hermitian.It is shown that if B is in (I) then x = By is a least-squares solution of Ax = y and that if B is in (2) then x = By is th e so lution of minimum Eu c lidian norm of the con sistent sys tems Ax = y.The connection is exposed bet wee n th e properties of the generalized inve rses in (a) an d (b) and the fa ct that among all matrices X satisfying AXA = A, that with minimum Euclidi an norm is the Moore-Penrose inverse of A.