## Evaluating the Fréchet derivative of the matrix $p$th root

João R. Cardoso

### Abstract

This paper shows that computing the Fréchet derivative of the matrix $p$th root is equivalent to solve a sequence of $p$ Sylvester equations. This provides the theoretical support to design an algorithm for the effective computation of the Fréchet derivative. The conditioning of the Sylvester sequence is addressed and some numerical experiments are carried out to illustrate the results.

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### Key words

conditioning, matrix $p$th root, Sylvester equation, Fréchet derivative, Kronecker products, perturbation theory

### AMS subject classifications

 Vol. 39 (2012), pp. 414-436 João R. Cardoso: Computation of the matrix $p$th root and its Fréchet derivative by integrals