Generalized Multiple Operator Integrals for Operators with Finite Dimensions

Multiple Operator Integrals (MOIs) have played a foundational role in operator theory and functional calculus, particularly for analyzing Hermitian matrices via spectral decomposition. Conventional MOIs rely on the assumption of self-adjointness, making them analytically tractable for computing Frechet derivatives, establishing trace formulas, and deriving commutator estimates. However, many problems in mathematics, science, and engineering involve matrices that are fundamentally non-Hermitian, arising in contexts such as numerical discretization of differential operators, signal processing, control systems, and non-Hermitian physics. These cases necessitate a more general framework for operator integration. In this paper, we propose and develop the theory of Generalized Multiple Operator Integrals (GMOIs), which extends MOI techniques to arbitrary matrices, including non-Hermitian and non-normal cases. We unify conventional MOIs under the perspective of the Spectral Mapping Theorem and present a rigorous construction of Generalized Double Operator Integrals (GDOIs) in finite-dimensional setting via Jordan decomposition of any matrix. We establish key algebraic properties, derive norm estimates, and prove continuity and Lipschitz-type perturbation formulas. Finally, we demonstrate how GMOIs can be used to compute matrix function derivatives in non-Hermitian environments, thus significantly broadening the applicability of MOI-based methods in both theoretical and practical domains.