Commutator Calculus and Symbolic Differentiation of Matrix Functions

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.