Real representations of powers of real matrices and its applications.

We give real representations of $ A^n $ A n (or $ e^{t A} $ e tA ) based on $ A^nP_{\mu } $ A n P μ for a real square matrix A, where $ P_{\mu } $ P μ is the projection to the generalized eigenspace associated with an imaginary eigenvalue μ of A. Our method is based on the spectral decomposition theorem. As applications, we can easily obtain realifications of representations of solutions of inhomogeneous linear difference equations with constant coefficients. [ABSTRACT FROM AUTHOR]