I want to compute the square root of a real symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ such that $S^{1/2}S^{1/2}=S$ and it's well known that this decomposition is unique.
My question is if I have $S$ a real matrix will its square root be real too or it could be a complex matrix. In case that it could be complex then $S$ could have infinitely many square roots.
Any comments?
Best Answer
A real matrix $S$ can possess infinitely many real or nonreal square roots. For example, $$ S=I=\begin{pmatrix}1&t\\0&-1\end{pmatrix}^2 $$ for every $t\in\mathbb{C}$. Note that $S=I$ is real and positive definite.