A Taylor expansion of the square root matrix functional

TL;DR

This paper derives explicit formulas for the derivatives of the matrix square root functional at any order, enabling Taylor expansions with integral remainders, which advances theoretical understanding of matrix functions.

Contribution

It introduces a novel formulation for computing higher-order Fréchet derivatives of the matrix square root functional and provides the first Taylor expansion of this functional with an integral remainder.

Findings

Explicit formulas for all orders of derivatives

First Taylor expansion with integral remainder for the matrix square root

Enables systematic computation of derivatives for matrix functions

Abstract

This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of the matrix square root at any order starting from the first order derivative. A Taylor expansion at any order with an integral remainder term is also provided, yielding the first result of this type for this class of matrix functional.