Eigenvalues and Normalised Eigenvectors

[captainjack2000]

Homework Statement

I have a matrix
H= [h g
g h]
and I need to find the eigenvalues and normalised eigenvectors

Homework Equations

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?

 

[radou]

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.

 

[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?

 

[captainjack2000]

are these normalised?

 

[radou]

What does it mean for a vector to be normalized?

 

[captainjack2000]

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

 

[radou]

I am not going to check your calculations, but with respect to the standard norm neither (1, 1) nor (1, -1) have unit length, since |(1, 1)|=(|(1, -1)|) = √2.