How to find $\sigma^2$ for $\sigma^2= \sigma_x^2 + \sigma_y^2 + \sigma_z^2$ where $\sigma_x^2$, $\sigma_y^2$ and $\sigma_z^2$ are the identity matrix?
My attempt:
$\sigma^2 = 3 {\bf I}$ (identity matrix) so can I clarify that since only the number 3 lies diagonally in the matrix, its eigenvalue is 3? Is it correct?
For another case where $\sigma = \sigma_x + \sigma_y + \sigma_z$ where 
How to find the eigenvalue in this case?
My attempt:
I realized when I add them up together, it isn't a diagonal matrix so how to find its eigenvalue in this case?
Use the definition of eigenvalue: the eigenvalues of a matrix $M$ are the solutions $\lambda$ of $$|M-\lambda I|=0$$
This definition works for diagonal and non-diagonal matrices. In the specific case of diagonal matrices, the eigenvalues are indeed the diagonal elements, so the eigenvalues of $3I$ are indeed all $3$.