## Diagonalizing hermitian matrices - how to get eigenvectors after finding eigenvalues?

1. The problem statement, all variables and given/known data
I can find my eigenvalues just fine, and they're both real, as expected. My first eigenvalue is -3, which I know is correct.

I have the equations 5x+(3-i)y=0, (3+i)x+2y=0

Both of the equations come from my hermitian matrix, after I substituted λ=-3.

2. Relevant equations

3. The attempt at a solution

I have absolutely no idea how to solve this. This case is simple enough to be solved by trial and error, but how would I proceed if I had harder equations?

I can't use both equations since I get x=x or y=y if I substitute one into the other, since they're both the same equation.
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 Nevermind, doing another (easier) exercise allowed me to see that I only have to set ax = by, and force a value to either x or y.