Letters to the Editor: Chaganty, N. R. (1993), "Comment," the American Statistician, 47,158: Comments by Bancroft and Neudecker and Liu with response by Chaganty and Vaish

Chaganty (1993), in a letter to the editor commenting on an article by Olkin (1992), offered as a theorem a version of the Cauchy-Schwartz inequality. When B is a symmetric nonnegative definite matrix with Moore-Penrose inverse B+, it has a symmetric nonnegative definite square root C with Moore-Penrose inverse C+. This follows from the singular value decomposition of B. The matrices BB', B+B, CC', and C+C are all equal to PB, the projection matrix into M(B), the column space of B. Apply the Cauchy-Schwartz inequality to the n x 1 vectors Cx', C+b' to obtain.