# Eigenvalues and Normalised Eigenvectors

1. May 4, 2010

### captainjack2000

1. The problem statement, all variables and given/known data
I have a matrix
H= [h g
g h]
and I need to find the eigenvalues and normalised eigenvectors

2. Relevant equations

3. The attempt at a solution
I subtracted lamda from the diagonal and then solved for the determinant equally zero. The eigenvalues I found were
(h-lambda)^2=g^2
so (h-lambda)=+/- g
lamdba=h+/-g

but I'm not sure how to find the normalised eigenvectors?

2. May 4, 2010

After finding the eigenvalues, plug them back into the equation (A - λ I)x = 0, one by one, to get your eigenvectors. Then normalize them.

3. May 4, 2010

### captainjack2000

Thank you very much.

So what I have done is
for eigenvalue h+g I have two equations
hx+gy=hx+gx and gx+hy = hy+gy which gives x=y so eigenvalue h+g has eigenvector (1,1)
and for eigenvalue h-g I have two equations
hx+gy=hx-gx and hx+gy=hy-gy so x=-y so eigenvalue h-g has eigenvector (1,-1)

Is that correct?

4. May 4, 2010

### captainjack2000

are these normalised?

5. May 5, 2010

What does it mean for a vector to be normalized?

6. May 5, 2010

### captainjack2000

to be of unit length...which these are right?

7. May 5, 2010