1. The problem statement, all variables and given/known data i have matrix A which is diagonalisable by doing an example on wiki under the section " how do diagonalise a matrix" http://en.wikipedia.org/wiki/Diagonalizable_matrix i realise that A x is not equal to [itex]\lambda[/itex] x , where x are eigen vectors of A , [itex]\lambda[/itex] is eigen values instead x A = [itex]\lambda[/itex] x and i think A x = x [itex]\lambda[/itex] QUESTION 1) if this is so, why do they always write the hamiltonian as H |x> = E |x> ??? shouldn't it be H |x> = |x> E ? if i remembered correctly, for matrix multiplication AB =/= BA right? but i read wiki and it says something like (i can't remember the exact phrasing) "it is equal if both A and B are diagonalisable matrix , and are both n by n matrix. " QUESTION 2) also, for PT A P = [itex]\lambda[/itex] if i want to "bring over" the P, is it like this A = P [itex]\lambda[/itex] PT but why is it like this? thanks!