Frechet derivatives of the power function
Rajendra Bhatia, John Holbrook · Indiana University Mathematics Journal · 2000
A → A r be the map that takes a positive definite matrix to its r th power, and let DA r be the Fréchet derivative of this map.It is known that DA r = r A r -1 if r ≤ 1 or if r ≥ 2, and also that this is not so if 1 < r < √ 2. We show that this equality between norms is true also when √ 2 ≤ r < 2, and thus resolve this question completely.Motivation for such results comes from the need for perturbation bounds on the power functions; we discuss this application..