Eigenvector of Pauli Matrix (z-component of Pauli matrix)

[roshan2004]

I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using $$(\sigma _z-\lambda I)X=0$$,
I got $$\left( \begin{matrix} 0 & 0 \\ 0 & -2 \end{matrix} \right)\left( \begin{matrix} x \\ y\end{matrix} \right)=0$$
Now, how can I find the eigen vector for eigen value 1 with this relation since it will give me only $$-2y=0$$

 

[HallsofIvy]

Yes, that says that y= 0. Since you have NO equation restricting x, x can be any thing. An eigenvector corresponding to eigenvalue 1 is (x, 0)= x(1, 0).

(Perhaps you are missing the fact that the set of all eigenvectors corresponding to a single eigenvalue is a subspace of the vector space. The set of eigenvectors corresponding to eigenvalue 1 is the one dimensional subspace of all multiples of (1, 0).)

 

[fzero]

That equation is solved by $y=0$; $x$ can be anything, but you can determine the normalized eigenvector if you want a concrete answer.