rtist
The largest eigenvalue is one way to summarize the whole matrix into one number, like the determinant. But determinant is zero, if the matrix is singular.
However, the largest root is often too variable. There are other measures to choose from, e.g., the sum of all eigenvalues, the sum of all eigenvalues/(1+eigenvalues), the product of 1/(1+eigenvalues), ......
OOPS, I just found that you are asking the eigenvectors :P