Abstract
In this note we present a simple proof of a theorem allowing us to castf(Â), where  is a non-singular matrix andf a function admitting a McLaurin expansion, as a finite sum. We also discuss the complementary version of the theorem and, limiting ourselves to 2 × 2 and 3×3 matrices, we show how they can be cast in an exponential form. Such form greatly simplifies the task of finding the π-th power (with π being any real or complex number) of a given matrix. The applications to physical problems like the optical-resonator stability and the Cabibbo-Kobayashi-Maskawa matrix are also discussed.
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