1.5. The Discrete-Time Case: Linear Difference Equations 19 Jordan form presented in Theorem 1.2.3 provides the following analogues to Propositions 1.3.1 and 1.3.2. Theorem 1.5.1. Let A ∈ Gl(d, R) with real generalized eigenspaces Ek = E(μk)k = 1,...,r, for the eigenvalues μ1,...,μr ∈ C with nk = dim Ek. (i) If A = T −1 JRT , then An = T −1 ( JR )n T , i.e., for the computation of powers of matrices it is suﬃcient to know the powers of Jordan form matrices. (ii) Let v1,...,vd be a basis of Rd, e.g., consisting of generalized real eigenvectors of A. If x0 = ∑ d i=1 αivi, then ϕ(n, x0) = ∑ d i=1 αiϕ(n, vi) for all n ∈ Z. (iii) Each real generalized eigenspace Ek is invariant for the linear dif- ference equation xn = Anx0, i.e., for x0 ∈ Ek it holds that the corresponding solution satisfies ϕ(n, x0) ∈ Ek for all n ∈ Z. This theorem shows that for explicit solution formulas iterates of Jordan blocks have to be computed. We consider the cases of real and complex eigenvalues. Example 1.5.2. Let J be a Jordan block of dimension m associated with the real eigenvalue μ of a matrix A ∈ gl(d, R). Then J = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ μ 1 · · · · · · · 1 0 μ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = μ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 · · · · · · · 0 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥+⎢ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 · · · · · · · 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Thus J has the form J = μI + N with N m = 0, hence N is a nilpotent matrix. Then one computes for n ∈ N, Jn = (μI + N)n = n i=0 n i μn−iN i . Note that for n ≥ m − 1, (1.5.3) ϕ(n, y0) = Jny0 = μn+1−m m−1 i=0 n i μm−1−iN i y0. For y0 = [y1,...,ym] the j-th component of the solution ϕ(n, y0) of yn+1 = Jyn reads ϕj(n, y0) = n 0 μny j + n 1 μn−1y j+1 + . . . + n m − j μn−(m−j)y n .

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