Why is the Hamiltonian written as H|x> = E|x> instead of H|x> = |x>E?

[quietrain]

Homework Statement

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 realize 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 P^T A P = [itex]\lambda[/itex]

if i want to "bring over" the P, is it like this

A = P [itex]\lambda[/itex] P^T

but why is it like this?



thanks!

 

[niklaus]

You are right that matrices don't commute, however the multiplication of a matrix with a scalar (real or complex number) always commutes. So λx = xλ. What you write above is also not correct. Ax is not at all the same as xA, the wiki page is correct.

Question 2):

this works for so called orthogonal matrices where P^T * P = P*P^T = 1 (the identity matrix). It is not true for general matrices.

 

[HallsofIvy]

You do not have to choose orthonormal vectors for eigenvalues but you can. If you do then the matrix P is "orthogonal"- in particular [itex]P^T= P^{-1}[/itex]. Without P being orthogonal, that is not true but it is still true that [itex]P^{-1}AP= D[/itex] so, multiplying on the left by P and on the right by [itex]P^{-1}[/itex],
[tex]P(P^{-1}AP)P^{-1}= PDP^{-1}[/tex]
[tex](PP^{-1})A(PP^{-1})= IAI= A= PDP^{-1}[/tex]

In the case that P is "orthogonal", You can replace [itex]P^{-1}[/itex] with [itex]P^T[/itex].

 

[quietrain]

oh shucks ... for question 1 i realize i multiplied the P ,matrix of eigen vectors, with the diagonal matrix eigen values

but in fact i think they are talking about the individual eigen value and eigen functions , not the combined P.

so yup i get it thanks everyolne!