bibkey: bathory2025commutator
authors: Michal Bathory
year: 2025
title: "Commutator calculus and symbolic differentiation of matrix functions"
doi: 10.1137/25M1760209
url: https://arxiv.org/abs/2505.07987v3
claim: "The paper develops a calculus of functions of the left and right multiplication operators of a real symmetric matrix, expresses derivatives of matrix functions through the anticommutator and commutator operators, proves identities such as ad_G = (log L_A − log R_A)/2 for A = e^G, and conjectures that, with a suitable generalization of the defining functional calculus, these identities hold for every complex square matrix G."
strata_touched:
  - D5/S3/Quantum/Matrix/CommutatorLogarithmBranchRefutation
license: citation-only
triage: anchor

Commutator calculus and symbolic differentiation of matrix functions

M. Bathory, arXiv:2505.07987v3; SIAM J. Matrix Anal. Appl. (2026). Quotations are from the v3 TeX source.

The operators:

\ml_{\mathbf{G}}\mathbf{X}:=\mathbf{G}\mathbf{X},;;\mr_{\mathbf{G}}\mathbf{X}:=\mathbf{X}\mathbf{G},;;\ac_{\mathbf{G}}\mathbf{X}:=\frac{\mathbf{G}\mathbf{X}+\mathbf{X}\mathbf{G}}2,;;\ad_{\mathbf{G}}\mathbf{X}:=\frac{\mathbf{G}\mathbf{X}-\mathbf{X}\mathbf{G}}2,

Lemma 3.4:

Let $p,q\in \mathbb{R}$, $\mathbf{G}\in {\mathbb{R}}_{\mathrm{sym}}^{d\times d}$ and set $\mathbf{A}=e^{\mathbf{G}}$. Then

\ad_{\mathbf{G}}&=\frac12(\log\ml_{\mathbf{A}}-\log\mr_{\mathbf{A}}).\label{al6}

Conjecture 3.7:

With a~suitable generalization of Definition~\ref{Def}, all identities of Theorem~\ref{Tform}, Lemma~\ref{Had}, and Corollaries~\ref{corpow},~\ref{Cor} (except \eqref{eq1}, \eqref{eq2}) continue to hold for every $\mathbf{G}\in {\mathbb{C}}^{d\times d}$ and for any $\mathbf{A}\in {\mathbb{C}}^{d\times d}$ such that $\log \mathbf{A}$ exists.

Verified locator

Source: https://arxiv.org/abs/2505.07987v3

• URL: https://arxiv.org/abs/2505.07987v3 (source cc_arxiv_1_.tex, md5 3f5e6f983ebd4bc8e0036ae4a4b880db):
  • the operators, Eq. (1.1), l. 38;
  • Lemma 3.4, l. 377–387, with Eq. (3.42) at l. 385;
  • Conjecture 3.7, l. 505–507.
• DOI: 10.1137/25M1760209 (https://doi.org/10.1137/25M1760209).