Commutator calculus and symbolic differentiation of matrix functions

TL;DR

This paper introduces a new functional calculus for applying functions to matrix commutator and anti-commutator operators, providing explicit symbolic derivatives useful in mechanics and PDE theory.

Contribution

It develops a coordinate-free calculus for matrix functions that simplifies derivative calculations and extends existing formulas like Daleckii--Krein to symmetric matrices.

Findings

Provides explicit symbolic derivative formulas for matrix functions.

Applies calculus to continuum mechanics and PDEs.

Simplifies derivative computations in matrix analysis.

Abstract

We propose a functional calculus which allows one to apply functions to the matrix anti-commutator/commutator operator. The calculus is introduced in a straightforward manner if the operators act on symmetric matrices, and it leads to a coordinate-free version of Daleckii--Krein formula. In this sense, the proposed calculus provides symbolic formulae for the derivatives of matrix-valued functions that are explicit and easy to use. We discuss several applications of the newly introduced calculus in continuum mechanics (Hencky logarithmic strain, objective rates, spin tensors, viscoelastic fluids) and in the theory of partial differential equations.