1. The problem statement, all variables and given/known data There is an Hamiltonian operation which is given by (2 1 1) (1 2 1) = H ; 3-by-3 matrix (1 1 2) And let's have an arbtrary eigenvector (a) (b) = v ; (3x1) matrix (c) Then, from the characteristic equation, the eigenvalues are 1,4. Here eigenvalue 1 is repeated one. 2. Relevant equations Now, my question arises. I know that the eigenvectors that corresponds to eigenvalue 1 is two and both are orthgonal to each other. However, I can't find any of them because when I substitute eigenvalue into 1, I get a kind of meaningless(?) equation, (2 1 1) (a) (a) (1 2 1) (b) = (b) (1 1 2) (c) (c) . And it gives 3-equivalent equation, a+b+c = 0 I couldn't determine any relation between a,b,c. 3. The attempt at a solution So I quess that do I have to choose a,b,c satisfying the condition a+b+c=0 but still problem doesn't disappear beacuse there would be a lot of soulutions.... What did I wrong?????