I am currently writing a program to find the 3 eigenvectors of a 3 by 3 matrix. I am only dealing with matrix with real elements and real eigenvalues. I know how to find the eigenvalues. However, I don't know how to find the eigenvectors.

Of course I know that

(Original Matrix) (Eigenvector) = (Eigenvalue) (Eigenvector)

But I don't how to solve this equation to find the eigenvectors.

Is there any method or routine or equations I can use to find the eigenvectors?