What is it about?
This work concerns a generalisation of standard functions, where real numbers are replaced by symmetric matrices. When differentiating such functions, difficulties arise due to the non-commutativity of matrix multiplication. This issue is addressed here by proposing new differentiation rules involving the matrix commutator operator. These rules are then applied to problems in PDE theory and continuum mechanics.
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Why is it important?
Symmetric tensors and their associated functions arise in many areas of physics, such as classical and quantum mechanics. Not only is it easier to differentiate these functions directly and symbolically, rather than relying on conditional series expansions, but this approach also provides a useful tool for obtaining general results in the theory of matrix functions.
Perspectives
I realized that the initial idea for this work dates back to September 2023. At that time, I was attempting a change of variables in the viscoelastic Oldroyd-B model that would lead to a more manageable equation.
PhD Michal Bathory
Charles University
Read the Original
This page is a summary of: Commutator Calculus and Symbolic Differentiation of Matrix Functions, SIAM Journal on Matrix Analysis and Applications, February 2026, Society for Industrial & Applied Mathematics (SIAM),
DOI: 10.1137/25m1760209.
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