This paper systematically establishes a complete extension of Operational Mathematics to matrix algebras. The core idea—extending the number of repetitions of basic mathematical operations from natural numbers to integers, rational numbers,real numbers,and complex numbers—is carried over to the non-commutative setting of n×n complex matrices. Simultaneously,matrix calculus (differentiation and integration) and matrix calculus of variations (variation and inverse variation) are shown to be natural manifestations of this extension in the continuous case.We propose a complete axiomatic system tailored to matrix algebras,rigorously define integer-order, fractional-order, real-order,and complex-order iterations for matrix functions, prove existence at each level using matrix Schr¨oder equations,matrix Abel equations, a matrix Kneser construction, and transfinite induction, and establish uniqueness theorems under regularity conditions that respect the spectral properties of matrices. We deeply explore the singularity structure of complex order matrix iterations and their connection with complex dynamical systems on matrix spaces. Furthermore,we prove that matrix fractional calculus and matrix fractional calculus of variations are special cases of Operational Mathematics in the continuous setting, thereby unifying discrete hyperoperations and continuous analysis within a single theoretical framework that respects non-commutativity. This paper further reveals a profound duality between the mathematics of numbers and the mathematics of operations,which extends naturally to matrix algebras, and transforms all open problems into rigorously proven theorems, laying a solid theoretical foundation for Operational Mathematics in non-commutative settings.
shifa liu (Wed,) studied this question.
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