Rounding errors leading to wrong eigenvector calculation

Rounding errors leading to wrong eigenvector calculation

Steps involved:
1. $\det{A - \lambda I} = 0$, solve for $\lambda$, the eigenvalues, say $\lambda = l$
2. $(A - lI)x = 0$, solve for $x$, the eigenvectors
We want the matrix $A- \lambda I$ to be singular, but when there are rounding errors, this will not be true anymore. The above example illustrates this problem.