Start with the eigenvalue equation (Amathbf{v}=lambdamathbf{v})
Rewrite it as ((A-lambda I)mathbf{v}=0)
Substitute the given eigenvalue (lambda) into (A-lambda I)
Solve the homogeneous system ((A-lambda I)mathbf{v}=0)
Find the null space of (A-lambda I)
Choose any nonzero vector in that null space as an eigenvector
If needed, express the eigenvectors as a basis for the eigenspace
Verify by checking that (Amathbf{v}=lambdamathbf{v})