An alternative proof on higher order derivatives of a multilinear map

Abstract

As a generalization of the formulas proved by Bhatia, Grover and Jain (Derivatives of tensor powers and their norms. Electron J Linear Algebra. 2013;26:604–619), in recent papers (The kth derivative of the immannant and the χ-symmetric tensor power of an operator. Electron J Linear Algebra. 2014;27:Article 18, On derivatives and norms of generalized matrix functions and respective symmetric powers. Electron J Linear Algebra. 2015;30:Article 22) Carvalho and Freitas obtained formulas for directional derivatives, of all orders, for generalized matrix functions and for every symmetric tensor power associated with a character ξ of a subgroup G of the symmetric group . Throughout our work, we used some well-known formulas for the derivatives of all orders of a multilinear map, since the maps that we studied are all multilinear. In this paper, we intend to present an alternative proof of these formulas, using the multilinearity argument.