This paper uncovers a parametrisation of second-order commutative matrices, revealing its impact on differential equations in quantum physics.
For two square matrices [Formula: see text], the matrix [Formula: see text] is said to be second-order commutative with respect to [Formula: see text] if they are not commutative but [Formula: see text]. In this paper, the general explicit parametrisation of second-order commutative [Formula: see text] and [Formula: see text] pairs will be given. We also consider its applications in solving certain differential equations arising in an NMR model and to the parametrisations of ladder operators in quantum physics.
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Takeo Kamizawa (2025) studied this question.
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