# Eigenvectors of a 2D hermitian operator (general form)

1. Feb 7, 2012

### gboff21

1. The problem statement, all variables and given/known data
Calculate the eigenvectors and eigenvalues of the two-dimensional
matrix representation of the Hermitean operator $\hat{O}$
given by

|v_k'>$\left(O$|v_k>= {{O_11,O_12},{O_21,O_22}}

where all Oij are real. What does Hermiticity imply for the o -
diagonal elements O12 and O21? For the eigenvectors you may
assume that the two base vectors read

v_1= (1) v_2= (0)
........(0)...........(1)

2. Relevant equations
None

3. The attempt at a solution

So, I know how to work out eigenvectors/values:

|A-λI|=0

so, I end up with (O11-λ)(O22-λ)-O12O21=0 as the characteristic equation

with solution λ= 1/2(O22+O11)±1/2√((O22+O11)^2-4(O11O22-O21O12))

the matrix O, being hermitian, makes O12=O21*

but how do I progress on to an actual answer?

2. Feb 7, 2012

### vela

Staff Emeritus
Since the matrix is real, you know that O21* = O21. You should be able to show that
$$\lambda = \frac{(O_{11}+O_{22}) \pm \sqrt{(O_{11}-O_{22})^2 + 4O_{12}^2}}{2}$$ That's about as simple as you can get it. Now you want to solve for the eigenvectors. You just have to grind through the algebra.

3. Feb 7, 2012

### gboff21

But it just ends up as a horrible mess which cannot be the answer!

4. Feb 8, 2012

### vela

Staff Emeritus