How can the eigenvector corresponding to zero eigenvalue be found out? I was trying with the following simple matrix in Matlab:
$$A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 2 & -3 & 4 \\ 3 & -4 & 5 \end{array}\right] \; .$$
In matlab computations, the matrix seemed nearly singular with one of the eigenvalues very close to zero (3e-15). That means the usual shifted inverse power methods for finding out the unit eigenvector corresponding to an eigenvalue won't work. But Matlab returns an eigenvector corresponding to 0. How? Basically, I would like to develop a program to compute this eigenvector given any singular matrix. What algorithm should I use?
Edit: (1) Edited to reflect that the 'nearly singular' comment was corresponding to Matlab calculation.
(2) Edited to specify the actual question.
Best Answer
This matrix is singular, the determinant is zero, so it has an eigenvector for eigenvalue $0$. Nothing mysterious there -- you might want to check the calculation that made you think it was only nearly singular.
As for how to find eigenvectors with eigenvalue $0$: They are just the solutions of the homogeneous system of linear equations corresponding to this matrix, $Ax=0$, so you can use e.g. Gaussian elimination.