Directional derivatives of the singular values of matrices depending on several real parameters

Abstract

In this document I recapitulate some results by Hiriart-Urruty and Ye (1995) concerning the properties of differentiability and the existence of lateral directional derivatives of the multiple eigenvalues of a complex Hermitian matrix function of several real variables, where the eigenvalues are supposed in a decreasing order. Another version of these results was obtained by Ji-guang Sun (1988). This 2020 version has been written following a remark raised by Miloud Sadkane about the presence or not of the value 1/2 in the formulas that give the lateral directional derivatives of multiple singular values. Therefore, I have added Theorem 9. I have also written in more detail the relationship of the reviewed results with those in the bibliography. Finally, I have also put a result of mine (Corollary 11 in [1]) (joint with Armentia and Velasco) which interprets a Lippert's Theorem (2005).