Write the matrix as (A)
Form the characteristic equation (det(A-lambda I)=0)
Compute the determinant of (A-lambda I)
Expand the determinant into a polynomial in (lambda)
Solve the polynomial equation for (lambda)
The solutions (lambda) are the eigenvalues
For each eigenvalue (lambda), verify by checking ((A-lambda I)v=0) has a nonzero vector (v)