Linear algebra - eigenvalues and eigenvectors and hermitian

  • Thread starter SpiffyEh
  • Start date
  • #1
194
0

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
[tex]\lambda[/tex]_ 1 = 1
[tex]\lambda[/tex]_ 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
[tex]\lambda[/tex]_ 1 x_1 (x_1)^H =
[1 -i
i 1]

[tex]\lambda[/tex]_ 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
 

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Answers and Replies

  • #2
307
3
No picture?
 
  • #3
614
0
Did you forget the picture, I can't see any.
 
  • #4
194
0
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
 
  • #5
351
1
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.
 
  • #6
194
0
oh! that worked perfectly. Thank you, I would've never gotten that
 

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