In this note we present a simple proof of a theorem allowing us to cast f(Â), where Â is a non-singular matrix and f 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. © 1993 Società Italiana di Fisica.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)
Dattoli, G., Mari, C., & Torre, A. (1993). A simplified version of the Cayley-Hamilton theorem and exponential forms of the 2 × 2 and 3 × 3 matrices. Nuovo Cimento della Societa Italiana di Fisica B, 108(1), 61 - 68. https://doi.org/10.1007/BF02874340