So I have found $3$ eigenvectors: $E(1): (2,-1,1), E(2): (1,0,0), (0,0,1)$
Where $E(i)$ is just the eigenvalue. So how do I determine my change of basis matrix? In my textbook they just say the change of basis matrix is:
$(1,0,0)^T, (0,0,1)^T, (2,-1,1)^T$ but i want to know how do they determine the order of the columns?
order of the columns determines the order if the diagonal matrix. for example, if $U = \pmatrix{2 & 1 & 0\\-1&0&0\\1&0&1},$ then the diagonal matrix will be $D= \pmatrix{1&0&0\\0&2&0\\0&0&2}.$ if you permute the columns of, you will permute the diagonal in the same way. in other words calling the columns of the matrix $U$ as $u_1, u_2, u_3$ you are writing the three equations $Au_1 = u_1, Au_2 = 2u_2, Au_3 = u_3$ in matrix form as $$AU = UD$$ the permutations of the columns can also be written as $AUP = UPP^TDP$ for any permeation matrix $P.$
suppose we switch the first and the third columns of $U$ and obtain $U_1 = \pmatrix{0&1&2\\0&0&-1\\1&0&1}$ then the new diagonal matrix will be $D_1 = \pmatrix{2&0&0\\0&2&0\\0&0&1}$ and we will still have $$AU_1 = U_1D_1 $$