On bounds for scaled projections and pseudoinverses

Let X be a matrix of full column rank, and let D be a positive definite diagonal matrix. In a recent paper, Stewart considered the weighted pseudoinverse X†D = (XTDX)−1XTD and the associated oblique projection PD = XX†D, and gave bounds, independent of D, for the norms of these matrices. In this note, we answer a question he raised by showing that the bounds are computable.