On continuity of the Moore-Penrose and Drazin generalized inverses (Q1239207)

It is well known that the operation of taking the Moore-Penrose generalized inverse \(A^+\) of an \(m\times n\) matrix \(A\) is discontinuous at \(A\) unless \(A\) has full rank. This note shows that the definition of the Moore-Penrose generalized inverse is continuous, however, in the sense that if \(\hat A\) almost satisfies the four defining relations of the Moore-Penrose generalized inverse, then \(\hat A\) is close to \(A^+\). Norm estimates make precise what is meant by close. The Drazin generalized inverse is considered for the case when Index \((A) - 1\). Resuts similar to those for the Moore-Penrose inverse are derived. The description of the ``continuity'' of the definition of the Drazin generalized inverse for matrices of arbitrary index remains an open problem.