tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 3

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 3 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Computing Generalized Eigenvectors. Let A be an eigenvalue of multiplicity m. Let n be the smallest integer such that rank nullspace A AI m. Let Nk denote the number of eigenvalues of rank k. These have the value Nk rank nullspace A AI k rank nullspace A AI fc-1 . One can compute the generalized eigenvectors of a matrix by looping through the following three steps until all the the Nk are zero 1. Select the largest k for which Nk is positive. Find a generalized eigenvector xk of rank k which is linearly independent of all the generalized eigenvectors found thus far. 2. From xk generate the chain of eigenvectors x1 x2 . xk . Add this chain to the known generalized eigenvectors. 3. Decrement each positive Nk by one. Example Consider the matrix 1 1 1 A 2 1 1 . 3 2 4 The characteristic polynomial of the matrix is 1 A 1 1 x A 1 A 1 2 3 2 4 A 1 A 2 4 A 3 4 3 1 A 2 4 A 2 1 A A 2 3. 854 Thus we see that A 2 is an eigenvalue of multiplicity 3. A 2I is 1 1 1 A 2I 2 1 1 3 2 2 The rank of the nullspace space of A 2I is less than 3. I 0 0 0 A 2I 2 11 1 1 1 1 The rank of nullspace A 2I 2 is less than 3 as well so we have to take one more step. 0 0 0 A 2I 3 0 0 0 000 The rank of nullspace A 2I 3 is 3. Thus there are generalized eigenvectors of ranks 1 2 and 3. The generalized eigenvector of rank 3 satisfies A 2I 3X3 0 0 0 0 0 0 0 X3 0 000 We choose the solution 1 X3 d . 0 .