8 1. Autonomous Linear Differential and Difference Equations Theorem 1.2.3. For every real matrix A ∈ gl(d, R) there is an invertible real matrix S ∈ Gl(d, R) such that JR = S−1AS is a block diagonal matrix, JR = blockdiag(J1,...,Jl), with real Jordan blocks given for μ ∈ spec(A) ∩ R by (1.2.1) and for μ, ¯ = λ ± ıν ∈ spec(A),ν 0, by (1.2.2) Ji = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ λ −ν ν λ 1 0 0 1 . . . 0 0 λ −ν ν λ . . . . . . . . . . . λ −ν ν λ 1 0 0 1 0 0 λ −ν ν λ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Proof. The matrix A ∈ gl(d, R) ⊂ gl(d, C) defines a linear map F : Cd → Cd. Then one can for μ ∈ spec(F )∩R find a basis such that the restriction of F to the subspace for a Jordan block has the matrix representation (1.2.1). Hence it suﬃces to consider Jordan blocks for complex-conjugate pairs μ = ¯ in spec(A). First we show that the complex Jordan blocks for μ and ¯ have the same dimensions (with appropriate ordering). In fact, if there is S ∈ Gl(d, C) with JC = S−1AS, then the conjugate matrices satisfy JC = S−1AS = ¯−1A ¯ Now uniqueness of the complex Jordan normal form implies that JC and JC are distinguished at most by the order of the blocks. If J is an m-dimensional Jordan block of the form (1.2.1) corresponding to the eigenvalue μ, then J is an m-dimensional Jordan block corresponding to ¯. Let zj = aj+ıbj ∈ Cm,j = 1,...,m, be the basis vectors corresponding to J with aj,bj ∈ Rm. This means that F (z1) has the coordinate μ with respect to z1 and, for j = 2,...,m, the image F (zj) has the coordinate 1 with respect to zj−1 and μ with respect to zj all other coordinates vanish. Thus F (z1) = μz1 and F (zj) = zj−1 + μzj for j = 2,...,m. Then F (z1) = Az1 = Az1 = μ z1 and F (zj) = Azj = Azj = zj−1 + μ zj, j = 2,...,m, imply that z1,..., zm is a basis corresponding to J. Define for j = 1,...,m, xj = 1 √ 2 (zj + zj) = √ 2 aj ∈ Rm and yj = 1 ı √ 2 (zj − zj) = √ 2 bj ∈ Rm.

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