The $k$-th derivatives of the immanant and the $\chi$-symmetric power of an operator

TL;DR

This paper extends formulas for derivatives of matrix functions to include immanants and symmetric powers, broadening the scope of previous derivative formulas for determinants and permanents.

Contribution

It generalizes existing derivative formulas to encompass immanants and symmetric powers, providing new tools for matrix analysis.

Findings

Derived formulas for derivatives of immanants and symmetric powers

Unified approach to derivatives of various matrix functions

Extended previous results on determinants and permanents

Abstract

In recent papers, R. Bhatia, T. Jain and P. Grover obtained formulas for directional derivatives, of all orders, of the determinant, the permanent, the $m$-th compound map and the $m$-th induced power map. In this paper we generalize these results for immanants and for other symmetric powers of a matrix.