Linear algebra - eigenvalues and eigenvectors and hermitian

[SpiffyEh]

Homework Statement

I attached the problem in a picture so its easier to see.

Homework Equations

The Attempt at a Solution

These are the values i got
\lambda_ 1 = 1
\lambda_ 2 = -1

x_1 = [-i; 1] (x_1)^H = [i 1]
x_2 = [ i; 1] (x_2)^H = [-i 1]
* where x_1 and x_2 are 2x1 matricies, and their hermitians are 1x2

after each multiplication I got
\lambda_ 1 x_1 (x_1)^H =
[1 -i
i 1]

\lambda_ 2 x_2 (x_2)^H =
[-1 -i
i -1]

When I add these together I get
[0 -2i
2i 0]

which is not the original A_5. I can't figure out where I went wrong in this process. If someone could look over it and let me know that would be great. Thank you

 

[Coto]

No picture?

 

[VeeEight]

Did you forget the picture, I can't see any.

 

[SpiffyEh]

I think I forgot to press the upload button after trying to attach it. Sorry about that. I edited it and it's there now

 

[hgfalling]

The spectral decomposition allows you to write a Hermitian matrix as a linear combination of projections onto the orthonormal basis of eigenvectors. So you need to normalize your eigenvectors, and then the 2 will nicely vanish.

 

[SpiffyEh]

oh! that worked perfectly. Thank you, I would've never gotten that