Start with a square matrix (A)
Find an eigenvalue (lambda) by solving (det(A-lambda I)=0)
Form the matrix (A-lambda I)
Solve ((A-lambda I)mathbf{v}=0)
Find a nonzero vector (mathbf{v}) in the null space of (A-lambda I)
Any nonzero scalar multiple of (mathbf{v}) is also an eigenvector
Repeat for each eigenvalue if needed