# Operator Reverse Monotonicity of the Inverse

Published in *American Mathematical Monthly*, 2011

I learned econometrics from Fumio Hayashi in 2006 using his textbook *Econometrics*. In an exercise related to deriving the variance matrix of efficient GMM on p. 245, Hayashi states without proof

Let \(A\) and \(B\) be two positive definite and symmetric matrices. \(A-B\) is positive semidefinite if and only if \(B^{-1}-A^{-1}\) is positive semidefinite.

I am a mathematically oriented person who is skeptical about mathematical statements until seeing their proofs. Although I found the proof in Horn & Johnson (1985) *Matrix Analysis*, Corollary 7.7.4(a), their proof was not very intuitive. In 2010, I came up with a very simple proof based on convex conjugate functions that work for any self-adjoint linear operators on Hilbert spaces.