UPPER BOUND AND STABILITY OF SCALED PSEUDOINVERSES

For given matrices X and D where D is positive definite diagonal, a we ighed pseudoinverse of X is defined by X(D)(+) = (X(H)D(2)X)+X(H)D(2) and an oblique projection of X is defined by P(D) = XX(D)(+). When X i s of full column rank, Stewart [3] and O'Leary [2] found sharp upper b ound of oblique projections P(D) which is independent of D, and an upp er bound of weighed pseudoinverse X(D)(+) by using the bound of P(D). In this paper we discuss the sharp upper bound of X(D)(+) over a set D -+ of positive diagonal matrices which does not depend on the upper bo und of P(D), and the stability of X(D)(+) over D-+.