1.2. The Real Jordan Form 9 Thus, up to a factor, these are the real and the imaginary parts of the generalized eigenvectors zj. Then one computes for j = 2,...,m, F (xj) = 1 √ 2 (F (zj) + F (zj)) = 1 √ 2 (μzj + ¯zj + zj−1 + zj−1) = 1 2 (μ + ¯) zj + zj √ 2 + ı 2 (μ − ¯) zj − zj ı √ 2 + xj−1 = (Re μ)xj − (Im μ)yj + xj−1 = λxj − νyj + xj−1, F (yj) = 1 ı √ 2 (F (zj) − F (zj)) = 1 ı √ 2 (μzj − ¯zj + zj−1 − zj−1) = 1 ı √ 2 (μ − ¯) 1 2 (zj + zj) + 1 2 (μ + ¯)(zj − zj) + yj−1 = (Im μ)xj + (Re μ)yj + yj−1 = νxj + λyj + yj−1. In the case j = 1 the last summands to the right are skipped. We may identify the vectors xj,yj ∈ Rm ⊂ Cm with elements of C2m by adding 0’s below and above, respectively. Then they form a basis of C2m, since they are obtained from z1,...,zm, z1,..., zm by an invertible transformation (every element of C2m is obtained as a linear combination of these real vectors with complex coeﬃcients). This shows that on the corresponding subspace the map F has the matrix representation (1.2.2) with respect to this basis. Thus the matrix A is similar over C to a matrix with blocks given by (1.2.1) and (1.2.2). Finally, Lemma 1.2.1 shows that it is also similar over R to this real matrix. Consider the basis of Rd corresponding to the real Jordan form JR. The real generalized eigenspace of a real eigenvalue μ ∈ R is the subspace gener- ated by the basis elements corresponding to the Jordan blocks for μ (This is ker(A − μI)n, where n is the dimension of the largest Jordan block for μ.) The real generalized eigenspace for a complex-conjugate pair of eigenvalues μ, ¯ is the subspace generated by the basis elements corresponding to the real Jordan blocks for μ, ¯ (See Exercise 1.6.9 for characterization which is independent of a basis.) Analogously we define real eigenspaces. Next we fix some notation. Definition 1.2.4. For A ∈ gl(d, R) let μk,k = 1,...,r1, be the distinct real eigenvalues and μk, μk,k = 1,...,r2, the distinct complex-conjugate eigenvalue pairs with r := r1 + 2r2 ≤ d. The real generalized eigenspaces are denoted by E(A, μk) ⊂ Rd or simply Ek for k = 1,...,r.

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