I have computed the eigenvalues and eigenvectors of a matrix numerically.
However, the matrix is relatively small and sparse, and I would like to go back and compute them manually, to state them analytically (as exact equations).
Here is the [A] matrix:
One of the eigenvalues is a REAL constant, k.
So, to solve for the associated eigenvector, the system becomes:
which is set to zero.
I have reduced this system to the following:
So the eigenvector can be stated in terms of the fourth variable:
HOWEVER, this result does not confirm the result I got for this eigenvalue numerically. The numerical result is complex; it has both real and imaginary elements, and all elements are non-zero. The analytic result I get above is real only.
Anybody here willing to verify my work, and point out where I went wrong?