6 1. Autonomous Linear Differential and Difference Equations Hence N may be chosen independent of A, B. Then the assertion follows, since there is δ 0 such that for A B δ, n=0 (An Bn) tn n! N n=0 (An Bn) tn n! + n=N+1 (An Bn) tn n! 2ε. 1.2. The Real Jordan Form The key to obtaining explicit solutions of linear, time-invariant differential equations ˙ = Ax are the eigenvalues, eigenvectors, and the real Jordan form of the matrix A. Here we show how to derive it from the well-known complex Jordan form: Recall that any matrix A gl(d, C) is similar over the complex numbers to a matrix in Jordan normal form, i.e., there is a matrix S Gl(d, C) such that JC = S−1AS has block diagonal form JC = blockdiag[J1,...,Js] with Jordan blocks Ji given with μ spec(A), the set of eigenvalues or the spectrum of A, by (1.2.1) Ji = μ 1 . . . 0 0 μ 1 . . . . . . . . . . . . 0 μ 1 0 0 μ . For an eigenvalue μ, the dimensions of the Jordan blocks (with appropriate ordering) are unique and for every m N the number of Jordan blocks of size equal to or less than m × m is determined by the dimension of the kernel ker(A μI)m. The complex generalized eigenspace of an eigenvalue μ C can be characterized as ker(A μI)n, where n is the dimension of the largest Jordan block for μ (thus it is determined by the associated map and independent of the matrix representation.) The space Cd is the direct sum of the generalized eigenspaces. Furthermore, the eigenspace of an eigenvalue μ is the subspace of all eigenvectors of μ its dimension is given by the number of Jordan blocks for μ. The subspace corresponding to a Jordan block (1.2.1) of size m × m intersects the eigenspace in the multiples of (1, 0,..., 0) Cm. We begin with the following lemma on similarity over R and over C. Lemma 1.2.1. Let A, B gl(d, R) and suppose that there is S Gl(d, C) with B = S−1AS. Then there is also a matrix T Gl(d, R) with B = T −1 AT.
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