Calculating eigenstates of Pauli matrices

Question

I need to find out the eigenvalues and the eigenstates of the Pauli matrices. I know that the eigenvalues of $\sigma_x$, $\sigma_y$, and $\sigma_z$ are all $\pm 1$. How do I find the eigenstates?

Answer 1

Eigenstates = eigenvectors.

To find the eigenvectors of a matrix $M$ for a given eigenvalue $\lambda$, you want to find a basis for the null space of $M - \lambda I$.

In your case, as each $M$ is $2 \times 2$ and you have two eigenvalues, the dimension of each eigenspace is $1$ and you are looking for one eigenvector for each eigenvalue.

For example, for $M = \sigma_z$ and $\lambda = 1$,

$$\sigma_3 - 1.I = \left( \begin{matrix} 0 & 0 \\ 0 & -2 \end{matrix} \right)$$

and the eigenvector $\displaystyle \left( \begin{matrix} 1 \\ 0 \end{matrix} \right)$, as this is a basis vector for the null space.