Real representations of powers of real matrices and its applications

We give real representations of A(n) (or e(tA)) based on A(n)P(mu) for a real square matrix A, where P-mu is the projection to the generalized eigenspace associated with an imaginary eigenvalue mu 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.