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

TL;DR

This paper reviews and clarifies mathematical results on the differentiability and directional derivatives of singular values of matrices depending on multiple real parameters, including recent corrections and interpretations.

Contribution

It provides a detailed review and correction of existing results on the directional derivatives of singular values, including new theorems and clarifications of their relationships.

Findings

Added Theorem 9 addressing the presence of 1/2 in derivative formulas

Clarified the relationship between eigenvalue and singular value derivatives

Included a new interpretation of Lippert's Theorem (2005)

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).