oo 8. ) Let ¢J(t,xo) be the solution of the initial value problem (1).

3vn + 2V2 + 3V3 + ... + (n VI + 2V2 + 3V3 + ... + (n - + (n - l)vn l)Vn - l + nvn . l)Vn - l Cf. Hirsch and Smale [HIS], p. 124. 8. Jordan Forms 43 Example 1. The only upper Jordan canonical forms for a 2 x 2 matrix with a real eigenvalue A of multiplicity 2 and the corresponding deficiency indices are given by [~~] and 81 = 1,82 = 2. Example 2. The (upper) Jordan canonical forms for a 3 x 3 matrix with a real eigenvalue A of multiplicity 3 and the corresponding deficiency indices are given by 1 0] A [o A 1 o 0 A 81 = 1,82 = 2,83 = 3.

N VI + 2V2 + 3V3 + ... + (n - + (n - l)vn l)Vn - l + nvn . l)Vn - l Cf. Hirsch and Smale [HIS], p. 124. 8. Jordan Forms 43 Example 1. The only upper Jordan canonical forms for a 2 x 2 matrix with a real eigenvalue A of multiplicity 2 and the corresponding deficiency indices are given by [~~] and 81 = 1,82 = 2. Example 2. The (upper) Jordan canonical forms for a 3 x 3 matrix with a real eigenvalue A of multiplicity 3 and the corresponding deficiency indices are given by 1 0] A [o A 1 o 0 A 81 = 1,82 = 2,83 = 3.