Let $A \in \mathcal{M}_{n,n} (\mathbb{R})$, with one eigenvalue $<0$ and one eigenvalue $>0$ (and there can be possibly more eigenvalues too, some of which may be zero as well). Can we find say about existence of a matrix $B \in \mathcal{M}_{n,n} ( \mathbb{R})$ , for which $B^2=A$ holds ?
(Here, $\mathcal{M}_{n,n} (\mathbb{R})$ denotes the set of all $n \times n$ matrices with all its entries $\in \mathbb{R}$.)