I have written a linear solver employing Householder reflections/transformations in ANSI C which solves Ax=b given A and b. I want to use it to find the eigenvector associated with an eigenvalue, like this:
(A-lambda*I)x = 0
The problem is that the 0 vector is always the solution that I get (before someone says it, yes I have the correct eigenvalue with 100% certainty).
Here's an example which pretty accurately illustrates the issue:
Given A-lambda*I
(example just happens to be Hermitian):
1 2 0 | 0 2 1 4 | 0 0 4 1 | 0
Householder reflections/transformation will yield something like this
# # # | 0 0 # # | 0 0 0 # | 0
Back substitution will find that solution is {0,0,0}
, obviously.