Commutator calculus and symbolic differentiation of matrix functions

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.

For an application to the logarithmic corotational derivative, see https://doi.org/10.48550/arXiv.2504.15692

References in corpus (4)

• Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
• Some remarks on the monotonicity of primary matrix functions on the set of symmetric matrices
• Viscoelasticity, logarithmic stresses, and tensorial transport equations
• Modeling metamaterials by second-order rate-type constitutive relations between only the macroscopic stress and strain