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Commutator calculus and symbolic differentiation of matrix functions

Michal Bathory

TL;DR

This work develops a coordinate-free functional calculus that applies functions to the matrix commutator and related operators, enabling symbolic differentiation of matrix-valued functions beyond power-series limitations. Central ideas include the left/right multiplication operators $\mathsf{L}_{\mathbf{G}}$, $\mathsf{R}_{\mathbf{G}}$ and the commutator/anticommutator operators $\mathsf{ad}_{\mathbf{G}}$, $\mathsf{ac}_{\mathbf{G}}$, along with a general Fréchet derivative formula $\frac{df(\mathbf{G})}{d\mathbf{G}}=\frac{f(\mathsf{L}_{\mathbf{G}})-f(\mathsf{R}_{\mathbf{G}})}{\mathsf{L}_{\mathbf{G}}-\mathsf{R}_{\mathbf{G}}}$. The paper provides explicit 2D/3D closed-form expressions for $f(\mathsf{ad}_{\mathbf{G}})$, derives comprehensive derivative identities for $\exp$, $\log$, powers, and trigonometric/hyperbolic functions, and establishes a DalekKi-Krein–type framework that supports symbolic manipulation without requiring spectral decompositions. It then demonstrates powerful applications to monotonicity, Sobolev-norm log-convexity, and inequalities, and derives a logarithmic reformulation of viscoelastic models (Oldroyd-B), linking logarithmic and upper-convected rates. Together, these results provide a rigorous, practical toolkit for basis-free analysis of matrix-valued PDEs and continuum mechanics models with noncommutative tensorial structure.

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.

Commutator calculus and symbolic differentiation of matrix functions

TL;DR

This work develops a coordinate-free functional calculus that applies functions to the matrix commutator and related operators, enabling symbolic differentiation of matrix-valued functions beyond power-series limitations. Central ideas include the left/right multiplication operators , and the commutator/anticommutator operators , , along with a general Fréchet derivative formula . The paper provides explicit 2D/3D closed-form expressions for , derives comprehensive derivative identities for , , powers, and trigonometric/hyperbolic functions, and establishes a DalekKi-Krein–type framework that supports symbolic manipulation without requiring spectral decompositions. It then demonstrates powerful applications to monotonicity, Sobolev-norm log-convexity, and inequalities, and derives a logarithmic reformulation of viscoelastic models (Oldroyd-B), linking logarithmic and upper-convected rates. Together, these results provide a rigorous, practical toolkit for basis-free analysis of matrix-valued PDEs and continuum mechanics models with noncommutative tensorial structure.

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.
Paper Structure (18 sections, 9 theorems, 134 equations)

This paper contains 18 sections, 9 theorems, 134 equations.

Table of Contents

  1. Motivation
  2. Functions applied to the commutator
  3. Rules of the commutator calculus
  4. Explicit formulae of function of the commutator in two and three dimensions
  5. Symbolic differentiation of matrix functions
  6. Formulae for derivatives of special matrix functions
  7. Further identities for the derivative of the matrix power and logarithm
  8. Possible extension to non-symmetric matrices
  9. Applications of the theory
  10. Monotonicity of matrix functions
  11. Log-convexity of the Sobolev norm for matrix-valued functions
  12. Inequalities involving matrix functions
  13. Logarithmic reformulation of the classical viscoelastic fluid models
  14. Connecting the logarithmic and upper-convected rates
  15. Comparing dissipation potentials
  16. ...and 3 more sections

Key Result

Theorem 2.2

Let $\mathbf{G}$, $\mathbf{g}$, $\mathbf{Q}$ and $U$ be as in Definition Def and let $f,f_1,f_2:U\times U\to\mathbb{R}$.

Theorems & Definitions (18)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • Corollary 3.6
  • ...and 8 more