So the textbook way is: 1. Roots of characteristic equation det(A-λI)=0 give the eigenvalues. 2. Solve (A-λI)x = 0 for each eigenvalue to find each eigenvecto

So the textbook way is: 1. Roots of characteristic equation det(A-λI)=0 give the eigenvalues. 2. Solve (A-λI)x = 0 for each eigenvalue to find each eigenvector x. It seems very plausible that step 2 can be reduced to an equation.