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) for a nonzero vector (mathbf{v})
Write the system of linear equations from ((A-lambda I)mathbf{v}=0)
Use row reduction to find the null space of (A-lambda I)
Express the solution vector (mathbf{v}) in parametric form
Choose any nonzero solution vector as an eigenvector
Repeat for each eigenvalue
Verify each eigenvector with (Amathbf{v}=lambdamathbf{v})