I have met a question to calculate the square root of a matrix in $\mathbb{R}^{s+1}$.
$$A = \left(\begin{array}{cc} \sigma_{11}^{2} & \sigma_{11}^{1/2}\Sigma_{1S}\Sigma_{SS}^{1/2}\\ \sigma_{11}^{1/2}\Sigma_{SS}^{1/2}\Sigma_{S1} & \Sigma_{SS}^{2} \end{array}\right)^{1/2},$$
where, $\sigma_{11}\in \mathbb{R},\Sigma_{1S}\in \mathbb{R}^{1*s},\Sigma_{S1}=\Sigma_{1S}^T, \Sigma_{SS}\in \mathbb{R}^{s*s}$ come from a covariance matrix $$B=\left(\begin{array}{cc} \sigma_{11} & \Sigma_{1S}\\ \Sigma_{S1} & \Sigma_{SS} \end{array}\right).$$ Can we directly write down the square root of $A^2$?
Or can we find the eigenvalues' relationship(or trace's relationship) between A and B?