Solving for eigenvalues in matrix equation. $H$ is a hermitian $4\times 4$ matrix, and after squaring twice I arrived at the following equation to solve:
$$(H^2-\alpha\Bbb1)^2=4\beta\Bbb1, \quad\text{where }\Bbb1 \text{ is the identity matrix and }\alpha,\beta \geq0.$$ 
According to the paper I'm reading the solutions for the eigenvalues of $H$ are: 
$$E^2_\pm=\alpha\,\pm 2\sqrt{\beta}$$
At least to get a similar form you could take a square root of the equation to get $$H^2=\left(\alpha\pm2\sqrt{\beta}\right)\cdot\Bbb1$$
Now I think this might be justified as follows: $H$ is hermitian so it's eigenvalues are real, hence the square of these must be positive or zero. $\alpha+2\sqrt{\beta}\geq0$ always, and we would only allow $\alpha-2\sqrt{\beta}$ if it is positive also. Although this doesn't really explain why other square roots of the identity matrix aren't allowed, like $-\Bbb1$.
The only extra information I believe is relevant is that $H$ isn't proportional to the identity matrix, but couldn't there in principle be repeated eigenvalues?
 A: The only eigenvalue of $(H^2 - \alpha I)^2$ is obviously $4\beta$. Now, if $A$ is a square matrix then the eigenvalues of $A^2$ are given by the squares of all eigenvalues of $A$. Therefore (going back), the only possible eigenvalues of $H^2 - \alpha I$ are $\pm 2\sqrt\beta$. But note that it could be only one of them. Now, again, if $A$ is a square matrix, then the eigenvalues of $A - \alpha I$ are given by $\lambda_i - \alpha$, where the $\lambda_i$ are the eigenvalues of $A$. Thus, the only possible eigenvalues of $H^2$ are $\alpha\pm 2\sqrt\beta$. These are two guys. If one of them is negative, it cannot be an eigenvalue of $H^2$. If both are positive, still only one could be an eigenvalue of $H^2$. Finally, the only possible eigenvalues of $H$ are $\pm\sqrt{\alpha\pm 2\sqrt\beta}$, depending on whether the square root exists.
A: Not exactly beautiful, but you know there are at most 4 eigenvalues and checking that all 4 of those work would suffice to show they must be of that form. Does the author assert that the matrix has distinct eigenvalues, or does the author just use the fact that they are of that form?
