- Thread Starter
- 17-01-2010 16:14
- 17-01-2010 16:55
For a general matrix A, suppose the eigenvectors are v1, v2, v3.
Then we know (1, 0, 0) can be written as a1v1+a2v2+a3v3 for some scalars v1, v2, v3.
Then multiplying by (A-x1I) "kills off" the a1v1 component (since A-x1I)v1 = 0.
Extending this idea, multiplying by (A-x1I)(A-x2I) kills off the a1v1 and a2v2 components, leaving only a multiply of v3, the third eigenvector.
[Note that there are various (uncommon) situations where this approach fails, either because v3 has the same eigenvalue as one of v1, v2 (in which case (A-x1I)(A-x2I) kills off v3 as well), or because the representation of (1, 0, 0) in terms of v1, v2, v3 has a3v3 = 0].