Perturbation theory for the matrix square root and matrix modulus

TL;DR

This paper derives first order perturbation formulas for the matrix square root and matrix modulus, extending existing differentiability results to include singular matrices.

Contribution

It introduces new first order perturbation formulas for the matrix square root and modulus, especially for singular matrices, expanding upon the Daleckii-Krein theorem.

Findings

New formulas for singular matrices

Extension of Fréchet differentiability results

Applicable to positive semi-definite and general matrices

Abstract

We provide first order perturbation formulas for the matrix square root (in the positive semi-definite case) and the matrix modulus (in the general case). The results are new for singular matrices, and extend previously known Fr\'{e}chet differentiability formulas provided by the Daleckii-Krein theorem.