I have a quick question about eigenvalues. If I'm given one eigenvector of a 3 by 3 matrix, I can easily calculate its corresponding eigenvalue.
I know I can determine the other two eigenvalues by factorising its characteristic polynomial. My question: Is there a way to work out the other two eigenvalues without factorising the characteristic polynomial? In other words, can I use the fact that the $det(P) = \lambda_1 \lambda_2 \lambda_3$ to work out the other two. For example if $\lambda_1 = 2$ and determinant is 20, $\lambda_2 \lambda_3 = 10$. This means the other two eigenvalues can either be 2 and 5 or 10 and 1.
Any useful insight would be much appreciated.